
Class _!XA^—~ 
Book >ks^ 



COPYRIGHT DEPOSm 



PRACTICAL MECHANICS 

• AND 

ALLIED SUBJECTS 



McGraw-Hill Book&mpaiiy 

tlGCtrical World TheLng^inGering and Mining Journal 
En^eGring Record Engineering News 

Railway Age G azette American Machinist 

Signal £ngirL(?Gr American Engineer 

Electric liailway Journal Coal Age 

Metallurgical and Chemical Engineering Power 






PRACTICAL MECHANICS 

AND 

ALLIED SUBJECTS 



JOSEPH- W<^L4>HALE, S. B., E. E. 

associate professor of engineering, the pennsylvania state 
college, detailed as supervisor apprentice schools, penn- 
sylvania lines east of pittsburgh and erie; member of 
the national association of corporation schools; 
member of the national society for the promo- 
tion of industrial education; associate 
member of the american institute of 
electrical engineers; author of 
"practical applied mathematics" 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 
239 WEST 39TH STREET, NEW YORK 

6 BOUVERIE STREET. LONDON, E. C. 

1915 



6^ 



Copyright, 1915, by the 
McGraw-Hill Book Company, Inc, 



0/.^ 



THE MAPIilC PRESS YORK PA 

©GI.A40f;7J5 4 

JUL 19 1915 



\^>^mr'^'Z 



/ 



PREFACE 

This book together with that on ^^ Practical Applied 
Mathematics^^ by the same author embodies the results 
of five years^ experience in the organization and develop- 
rfient of a system of railroad shop apprentice schools, as 
well as extended investigations of the work of public, 
private, trade, vocational and corporation schools of 
manufacturing industries and railroads in this country 
and abroad. 

In the presentation no formal distinction is made of the 
various branches of elementary mathematics and science 
involved. The subject matter is presented in the two 
volumes under the general titles of ^^ Practical Applied 
Mathematics'' and ^^ Practical Mechanics and Allied 
Subjects,'' although a careful and logical arrangement 
and sequence have been followed in the presentation. 

In organizing trade apprentice schools difficulty has 
been experienced in securing a text practically adapted 
to the needs of these schools. This difficulty has led the 
author to develop a series of instruction sheets which are 
believed to be thoroughly practical. This volume with 
that on ^^ Practical Applied Mathematics"- is essentially 
a presentation of problems arising in shop experience in 
the mechanical trades. It shows the use of mathematics 
as a tool. 

The material included in this work has been made 
sufficiently broad to apply not only to railroad schools 
but it is believed to schools in the mechanical trades 
generally. It is also felt that this work may be used to 



vi PREFACE 

advantage in the public schools in technical branches, as 
well as in trade and vocational schools, either as a regular 
or as a supplementary text. 

The material for each book is presented in twenty 
chapters, each dealing with a definite subject. This 
affords a flexibility highly desirable in assigning the work 
and enables it to be given entirely or in part according to 
specific needs. 

In preparing this work the author acknowledges 
indebtedness to Pennsylvania Railroad Motive Power 
Officials at Altoona. The author is indebted to Dr. J. P. 
Jackson, State Commissioner of Labor and Industry of 
Pennsylvania, Professor J. A. Moyer, Head of the 
Department of Mechanical Engineering and Director 
of the Engineering Extension Division of the Pennsyl- 
vania State College and Mr. M. B. King, State Expert 
Assistant in Industrial Education of Pennsylvania, for- 
reading the proofs and offering valuable criticisms and 
suggestions. He is especially indebted to Professor 
J. H. Yoder and Mr. E. W. Hughes of the School of 
Engineering of the Pennsylvania State College, detailed 
as Instructors in the Pennsylvania Railroad System of 
Apprentice Schools. They have assisted in the prepa- 
ration of the manuscript, carefully read the proofs and 
offered many valuable suggestions. 

Joseph W. L. Hale. 
Altoona, Pa, 
June 10, 1915 



CONTENTS 

Pace 

Preface . . , v 

CHAPTER I 

Forces 

Art. 

1. Definitions 1 

2. Forces passing through a point 2 

3. Parallel forces 2 

CHAPTER II 

Gravitation. Center of Gravity 

4. Definition of gravity 5 

5. Gravity action on freely falling objects 5 

6. Work done against gravity 6 

7. Definition of center of gravity . 7 

8. Stability of objects 12 

CHAPTER III 

Density and Specific Gravity 

9. Definitions and problems 14 

CHAPTER IV 

Screw Threads 

10. Thread sections 20 

11. Pitch and lead 20 

12. Single, double, and triple threads 21 

13. Right- and left-handed threads 21 

14. Construction of a single right-handed ''V thread. ... 22 

15. Construction of a double left-handed *'V" thread. ... 23 

vii 



viii CONTENTS 

Art. Page 

16. Construction of a United States Standard thread .... 23 

17. Construction of a single square thread 26 

18. Construction of an Acme thread "... 26 

19. Miscellaneous rules 28 

CHAPTER V 

Calculation of Levers 

20. Definitions and problems 31 

21. Calculation for levers taking into account the weight of the 
lever itself 36 

CHAPTER VI 
Pulleys. (Block and Tackle) 

22. Arrangement of pulley systems .... 48 

CHAPTER VII 
The Inclined Plane and Wedge. The Screw Jack 

23. The inclined plane and wedge 53 

24. The screw jack 56 

CHAPTER VIII 

Gears^ Lathe Gearing 

25. Definitions ... 59 

26. Gear trains. (Simple gearing) 60 

27. Gear Trains. (Compound gearing). ........ 60 

28. Lathe gears for screw cutting. (Simple gearing) .... 61 

29. Lathe gears for screw cutting. (Compound gearing) . . 64 

CHAPTER IX 
Belts and Pulleys. Efficiency of Machines 

30. Belts and pulleys 69 

31. Efficiency of machines 73 



CONTENTS IX 

CHAPTER X 
Motion 

Art. Page 

32. Definitions 75 

33. Composition of motions 76 

CHAPTER XI 

Cutting Speeds. Speeds of Lathes 

34. Definitions and problems 79 

CHAPTER XII 
Volume and Pressure of Gases 

35. Definitions and problems 87 

CHAPTER XIII 

Work and Power 

36. Definitions and rules on work 91 

37. Power. Horsepower 91 

CHAPTER XIV 

Calculation of Belting. Energy 

38. Horsepower of belting 100 

39. Definitions on energy and problems 102 

CHAPTER XV 

Heat 

40. Definitions 105 

41. Temperature 105 

42. Heat measurement 107 

43. Expansion and contraction due to heat 108 



X CONTENTS 

Art. Page 

44. Specific heat Ill 

45. Latent heat 112 

46. Mechanical equivalent of heat 112 

47. Heat produced by the electric current 112 

CHAPTER XVI 
Logarithms 

48. Definitions 116 

49. The characteristic 116 

50. The mantissa 119 

51. Use of the tables 119 

52. To find a number corresponding to a logarithm. . . . 120 

53. To raise a number to a power 122 

54. To find a root of a number 123 

CHAPTER XVII 

The Measurement of Right Triangles 

55. Definitions 127 

56. Right triangle rule . 127 

57. Definition of sine, cosine, tangent, and cotangent .... 129 

58. Tables of rules for functions of an angle 130 

59. Calculation of functions for 30° and 45° 130 

60. Explanation of use of tables 131 

61. Tables of sines, cosines, tangents, and cotangents. . . 133 

62. Line values of functions 157 

63. Value of the sine, cosine, tangent, and cotangent for the 
common angles 159 

64. Methods of working out right triangles 160 

65. Applications of triangular functions 167 

CHAPTER XVIII 

The Measurement of Oblique Triangles 

66. Methods of working out oblique triangles 177 

67. Applications of triangular functions to oblique triangles . 186 



CONTENTS 



CHAPTER XIX 
Electricity 

Art. Page 

68. Definitions and calculation of resistance 190 

69. Resistances in series 191 

70. Resistances in parallel 192 

71. Table of safe carrying capacity of wires 194 

72. Ohm's law. Calculation of current, voltage, and resist- 
ance 194 

73. Calculation of power and energy in an electric circuit . . .197 

74. Electrical horsepower 200 

75. Methods of charging for electrical energy 202 

76. Comparison of water and electric systems 204 

CHAPTER XX 

Strength of Materials 

77. Definition 208 

78. Stress and strain 209 

79. Kinds of stresses 209 

80. Ultimate strength 211 

81. Testing machines 212 

82. Standard test pieces 216 

83. Values of ultimate strengths 217 

84. Factor of safety 218 

85. The elastic limit 218 

86. Values of safe working stresses 219 

87. Strength of rods 220 

88. Strength of ropes, chains, and cables 221 

89. The strength of columns 222 

Index 225 



PRACTICAL MECHANICS 

AND 

ALLIED SUBJECTS 

CHAPTER I 
FORCES 

1. Definitions. — In our every day work we deal with 
forces of one kind or another. A pressure is a force. 
The earth exerts a force of attraction for all bodies or 
objects on its surface. Steam exerts a force or pressure 
in a locomotive boiler and cylinder. In grinding tools 
a force of friction has to be overcome. The wind exerts 
a force or pressure against the surface which it strikes. 

When forces act on objects they move the objects if the 
forces are great enough to overcome the resistance to mo- 
tion, and even if the objects do not move the forces pro- 
duce a stress and pressure within and on the objects. 

Forces are measured in pounds and tons. When we 
consider forces acting on objects we must know how the 
forces are applied, the direction of the forces and their 
values in pounds and tons. In working out problems 
forces are often represented by straight lines. The length 
of the line represents to scale the value of the force as 
100 lb., 20 tons, etc., and the direction of the line, the 
direction in which the force acts as vertical, horizontal, 
etc. 

1 



2 MECHANICS AND ALLIED SUBJECTS 

2. Forces Passing through a Point. — In Fig. 1 the 
line Od 1 in. in length represents a force of 15 lb. and 
the line Ob 2/3 of an in. in length represents a force of 
10 lb. Also force Ob is horizontal and force Od acts at 

an angle of 60° with the horizontal. 

J- -^^ Two or more forces may be combined 

^y 1^^^^ ^^ ^^^ ^^parallelogram'' method giv- 
V y^ ^/ ing a single force having the same effect 

/y%^ / as the separate forces acting together, 

^ IQ,^ '^ as for example in Fig. 1 forces of 10 

Yy^ \ lb. and 15 lb. acting on an object at 

and 60° apart may be replaced by 
a single force Oc of 22 lb. which is the same in effect 
as the forces of 10 lb. and 15 lb. acting together. The 
single force Oc is called the '^resulianV of the forces Od 
and Ob, In this Fig. lines Ob ^ 

and dc are. parallel and equal, q ^~ ^^^> c^?5-lb.^ 

and the same is true of lines ^""^^'f^- ^ 

Od and be. 

Rule. — The resultant of a number of forces is the direct 
sum of these forces only when they act in the same 
straight line, as in Fig. 2 and in the same direction. 

Any force has the greatest 
^fl^^^' ^'^k^^' ^ff^c^ ^"^ ^^^ direction in which 

^1 c J^ it acts, and no effect in a di- 

U — a — >K — b ?-• rection at right angles to that 

20 Z in which it acts. 

Fig. 3. 3. Parallel Forces. — Paral- 

lel forces are those which act 
in directions parallel to one another and which do not 
meet. In Fig. 3 is shown three forces, one of 20 tons 
and two of 10 tons each, balancing the load of 20 tons. 
For parallel forces as in this Fig. TT^i + W2 = 20, and 



FORCES 

Wi h_ 

20 ~ a + h 
and 

TF2 a 



20 a + 6 



PROBLEMS 

1. What is the total force or pressure acting on a 14-in. piston 
in a steam cyhnder with a steam pressure of 180 lb. per sq. in.? 

2. What must be the diameter of a piston in an air cylinder for 
a total pressure of 3000 lb. with a pressure per sq. in. of 60 lb.? 

3. If the total pressure produced on a piston by steam in an 
18-in. cylinder is 45,600 lb., what is the steam pressure in lb. per 
sq. in.? 

4. What force of attraction does the earth produce on a steel 
rod 4 in. in diam. and 3 ft. 6 in. in length? 

5. Two forces of 20 lb. and 30 lb. act on an object in opposite 
directions. What is the ''resultant" force? If these forces 
acted at right angles to one another what would be the ''resultant" 
force? 

6. What is the total pressure produced on the supporting 
structure for a water tank if the tank is 12 ft. high and 8 ft. inside 



100 T 
Fig. 4. 



diam. and full of water? The material of the tank weighs 1280 
lb. 

7. A traveling crane (Fig. 4) lifts a 100-ton locomotive. With 
the load half way between the crane supports A and 5, what pres- 
sure is brought on each of these supports? How is this pressure 
changed when the load c moves nearer end A ? 

8. Using Fig. 4, find the pressures produced on supports at 
both A and B with the following loads and distances of the loads 
from A and B. 



MECHANICS AND ALLIED SUBJECTS 





Load 
at C 


Distance 
AC 


Distance Pressure 
BC at A 


Pressure 
at B 


1 


20 tons 


20 ft. 


40 ft. 






2 


15 tons 


32 ft. 


28 ft. 






3 


100 tons 


10 ft. 


50 ft. 






4 


25001b. 


50 ft. 


10 ft. 






5 


30 tons 


42 ft. 


18 ft. 






6 


3901b. 


25 ft. 


35 ft. 







9. What steam pressure is used on locomotives? Is the same 
pressure used for heating cars and the shops ? 

Give complete answer with reasons. 

10. If the steam pressure is 30 lb. per sq. in., what is the total 
force tending to blow the end out of a steam pipe 4 in. inside 
diam.? 



CHAPTER II 
GRAVITATION. CENTER OF GRAVITY 

4. Definition of Gravity. — The attractive force which 
every object in the universe has on every other object is 
called ^^ gravitation^' and this depends on the quantity 
of material composing the objects. 

The force of attraction of the earth for objects is called 
the force of gravity and is what gives objects their ^^ weight.^ ^ 
A spring balance in giving the ^Sveighf of any object 
measures merely the force of attraction of the earth for 
that object. 

On account of 'Hhe force of gravity '' which tends to 
pull objects toward the center of the earth work has to 
be done in lifting them and for the same reason elevated 
objects have power to do work on account of their ele- 
vated position, as in the case of water above a fall, ele- 
vated weights, train on a down grade, lifted hammer, 
etc. A locomotive hauling a train up a grade consuiiies 
more coal than when the train is on level track since work 
has to be done in lifting the weight of the train against 
the force of gravity as well as overcoming frictional re- 
sistances. A train descending a grade has work done on 
it by the force of gravity. The train will coast and often 
attain a high speed without the use of any steam. 

5. Gravity Action On Freely Falling Objects. — The 
'^ force of gravity '^ acting on objects free to fall increases 
their speed about 32.2 ft. per sec. each second of the 

5 



6 MECHANICS AND ALLIED SUBJECTS 

fall. For example, a weight starting from rest and fall- 
ing freely will have a speed of 96.6 ft. per sec. at the end 
of the third second of its fall and 161.0 ft. per sec. at the 
end of the fifth second of its fall. 

6. Work Done Against Gravity. — The work done in 
lifting objects against ^Hhe force of gravity^' is measured 
in foot-pounds and is equal to the weight of the object 
in pounds multiplied by the vertical height in feet through 
which the object is lifted. 

PROBLEMS 

1. A train is made up of a locomotive weighing 272,000 lb., a 
158,000-lb. tender, a baggage and mail car weighing 86,000 lb. 
and 5 Pullman sleepers weighing together 625,000 lb. How 
much work is required to lift the train up a mile of grade rising 
1 ft. in 250 ft.? 

2. How much w^ork in foot-pouiids can a 2-ton hammer do in 
each fall if it is raised 16 ft.? 

3. How much work in foot-pounds is required to load the 
following lot of tires on a flat car, if the average distance through 
which they are raised is 6 ft.? 

50 tires each weighing 452 lb. 
37 tires each Aveighing 643 lb. 
24 tires each weighing 598 lb. 
16 tires each weighing 544 lb. 

4. If a 10-ton casting is raised 23 ft. by a traveling crane, how 
much work is done? If one horsepower is the performance of 
33,000 ft. -lb. of w^ork per min. and the above casting is raised in 
1 min., what horsepow^er does the motor on the crane exert? 

6. Using the definition for '^horsepower" given in Prob. 4, 
what work in foot-pounds could a 10-h.p. engine do in 5 min. if 
it ran at full capacity for this time? 

6. A single acting pump raises 1600 gal. of water every minute 
through a distance of 5 ft. How much work against gravity does 
the pump do per hr. ? 

7. A train weighing 1000 tons descends 2 miles of grade dropping 



GRAVITATION 7 

1 ft. in 200 ft. What work in foot-pounds does ''gravity" do on 
the train in the descent? 

8. How much work is done per hr. in raising a 1-ton hammer 
through a vertical height of 4 ft. an average of fifteen times per 
min.? 

9. If an object falls freely for 7 sec, what is its speed at the 
end of this time? Would a bullet and a feather fall side by side 
in the air? Has air friction any effect on the speed of a falling 
object? 

7. Definition of Center of Gravity. — In the fourth 
paragraph we learned that the earth attracts toward its 
center every particle of every object on its surface, and 
this force of attraction called the force of gravity gives 
objects their weight. When solving problems in mechan- 




/X 



Fig. 5. 



Fig. 6. 



ics it is of ten^ necessary to find a point in an object at 
which its entire weight may be considered as concen- 
trated. This point about which an object would bal- 
ance is called its center of gravity. For example, a circular 
steel plate as in Fig. 5, has its ^^ center of gravity'^ at 
the center 0, half way between the faces of the plate. If 
the metal in this plate weighed .30 lb per cu. in. and the 
plate contained 100 cu. in. of metal we could consider 
its entire weight of .30 X 100 or 30 lb. as acting at 0, 
instead of taking the weight of each cu. in. separately. 
This gives only one value, 30 to work with instead of 100 
values, each of .30. 



8 MECHANICS AND ALLIED SUBJECTS 

The entire weight of a square uniform metal* plate 
as in Fig. 6 may be taken at a point P at the inter- 
section of the diagonals of the square and half way be- 
tween the faces of the plate. The center of gravity of a 
cylinder is at a point on the axis of the cylinder and half 
way between the upper and lower bases. 

The position of the center of gravity of an object 

such as a metal sheet, as shown in Fig. 7, can be found 

experimentally by using a plumb line and a support on 

which the object can move freely. When an object is 

suspended from a support on which it can move freely the 

^^center of weight'^ or ^^cen- 
^ppor ^^^ ^£ g^^^-^yM jg brought 

as low as possible, and there- 
fore lies in a vertical line 
drawn through the point of 
support. Using this prin- 
ciple an object as a steel 
Fig. 7. plate shown in the figure 

is suspended freely from 
the support ^^s'^ and a plumb line dropped from the 
point of support as shown. The direction of the line is 
then marked on the plate giving line '' ^a^ 

The plate is next supported from a second point and 
with the plumb line we get line Siai. The point where 
lines sa and Si^i cross marks the position of the ^^ center 
of gravity '^ of the plate, half way between the front and 
back faces, if the plate is of uniform thickness throughout. 
The center of gravity of sections made up of a number 
of simple sections as shown in Fig. 8, composed in this 
case of three rectangles, is found as follows: First we 
choose two ^^ reference axes'' as lines ah and cd crossing 
at 0. Next we find the product of the area of each simple 




GRAVITATION 



9 



figure and the distance of its center of gravity from the 
chosen axis, then we add these products together and 
divide their sum by the sum of the areas of the simple 
figures. 

For example, in the figure shown, the area of rectangle 
number 1 is 2 X 13^^ or 3 sq. in. The area of No. 2 is 
6 X 3 or 18 sq. in. and of No. 3, 2 X 2 or 4 sq. in. The 



, - - Cen -fer of Oraviiy 



/ Cenhr of Orayj-ft/ 




Fig. 8. 



sum of the products of these areas by the distances of 
their respective centers of gravity from axis ah gives 
3 XI + 18 X 3 + 4 X 5 or 77. Dividing this by the 
sum of the areas of the simple figures we have 

77 77 ^^* 

3+T8"+4 ^^ 25 ^^ ^-0^ ^^- 

which is the distance of the center of gravity of the 
whole figure from line or axis ah. 

Working in the same way from axis cd we have the 
equation 



10 



MECHANICS AND ALLIED SUBJECTS 



Distance of e.g. from line 



cd = 



3 X .75+ 18 X 1.5 - 4 X 1 

~ 3+18 + 4 " 



25.25 , . , 

or ^r which equals 1.01 m. 



The point where the lines cross as indicated in Fig. 9 
is the position of the center of gravity of the figure or 
the point on which the whole plate would balance. In 
the equation above the last term in the numerator that 



^ 






















Co 


f6 




o 










h 


^ — 3:08 


"— > 









Fig. 9. 



is 4 X 1 has to be subtracted as shown rather than 
added since the area 4 sq. in. is on the other side of the 
reference axis cd, and if this was, for example, a metal 
plate we were calculating, the weight represented by 
the lower rectangle would help counterbalance the weight 
of the parts on the other side of the axis cd. 

Example. — Find the center of gravity of the metal 
plate shown in Fig. 10 in which are cut two holes 3 in 
and 6 in. in diam. 



GRAVITATION 



11 



Solution. — Distance of center of gravity from axis ab 
equals 

18 X 15 X 9 + 4 X 3 X 20 - 6'-^ X .785 X 7- 3^ X .785 X 14 
18 X 15 + 4 X 3 - 62 X .785 - 3^ X .785 

2373.3 
or ^.^ ' which equals 9.62 in. 
246.68 ^ 




9.6^"^ 



Fig. 10. 

Distance of center of gravity from axis cd equals 

18 X 15 X 7.5 + 4 X 3 X 1.5 - 6^ X .785 X 8-32X.785X8 
18 X 15 + 4 X 3 - 62 X .785 - 3^ X .785 

1873.4 



or 



246.68 



which equals 7.59 in. 



PROBLEMS 

10. Find the center of gravity of the angle iron section shown in 
Fig. 11. 

11. Find the center of gravity of the ^'Z" beam section shown 
in Fig. 12. If this beam is 10 ft. long, find its weight at .28 lb. 
per cu. in. for steel. 

12. Find the center of gravity of the I-beam section shown in 



12 



MECHANICS AND ALLIED SUBJECTS 



Fig. 13, and its weight if 18 ft. long and steel weighs .28 lb. per 
cu. in. 

8. Stability of Objects. — Whether an object will stand 
up or fall over, that is, its ^^ stability,'' depends upon 




T 



to' 






Q;> 



■"^ 
^ 



^J 



Fig. 13. 



the position of its ^^ center of gravity'' with respect to 
its base. The rule is as follows : If a vertical line through 
the ^^ center of gravity" passes inside the base of an ob- 
ject as in Fig. 14, the object will not fall, but if the 





Fig. 14. 



Figs. 15 (a) and (6). 



vertical line comes outside the base as in Fig. 15a, the 
object will fall. From this it follows that objects which 
are heavy and have a large base are less likely to fall, 
than those which are lighter and have a smaller base, 



GRAVITATION 13 

and also the lower the ^^ center of gravity '^ in an object 
the less likely it is to fall. The center of gravity of an 
'^area^^ or ^^ surface ^^ of any kind is understood to mean 
that point about which a very thin sheet representing 
the surface would balance. Fig. 156 representing the 
end of a locomotive shows the change in position of the 
^^ center of gravity'^ with respect to the base when the 
locomotive is on a depressed rail on a curve. 

PROBLEMS 

13. A steel plate 2 ft. X 4 ft. is % in. thick. What is its weight 
and at what point in the plate can this entire weight be considered 
as concentrated? What is this point called? 

14. Where is the ^^ center of gravity" of a sphere located? 

15. At what point in a steel bar 4 in. in diam. and 6 ft. long 
can the entire weight be considered as acting? What is this 
weight? 

16. Where is the center of gravity of a cube located? Find the 
weight and center of gravity of a cube of steel 2 in. on a side. 

17. Why is it desirable to have the center of gravity of a pulley 
in line with the center line of the shaft on which the pulley works? 

18. How could you find the center of gravity of any triangularly 
shaped piece of steel plate? 

19. Is it necessary to use a counterbalance on a fly-wheel, and 
if so explain why? 



CHAPTER III 
DENSITY AND SPECIFIC GRAVITY 

9. Definitions and Problems. — The density of an ob- 
ject means its mass or quantity of matter per unit of 
volume. Suppose a cubic foot of metal weighs 450 lb., 
we may then say that the density of the metal is 450 lb. 
per cu. ft. Since all objects are not of the same weight 
it is found convenient to have a ready means for calcu- 
lating their relative weights. In order to do this, tables 
are prepared which give us the weights of unit volumes 
of materials compared with the weight of the same vol- 
ume of some material taken as a standard. 

Specific Gravity is the ratio of the density of two ob- 
jects. For example: If a cubic foot of metal weighs 490 
lb. and it is compared with the weight of a cubic foot of 

490 
water which is about 62.5 lb., the ratio is tttt^ or 7.83 

oz.o 

which is the specific gravity of the metal referred to 

water. 

If a block of granite containing 1 cu. ft. volume 

weighs 162 lb., its weight compared to a cubic foot of 

162 
water is x^ or 2.59 which is its specific gravity referred 

to water as a standard. In the same way we can com- 
pare the weights of other materials with water and pre- 
pare a table which we can use to advantage in calculations 
involving the weights of the materials most used. 

In any standard handbook may be found such tables 

14 



DENSITY AND SPECIFIC GRAVITY 15 

already prepared for a large number of substances. In 
making out these tables the weights of the bodies if 
they are solid or liquid are compared to the weight of 
the same volume of water as a standard, the weight of 
the water being taken as that at its temperature of 
greatest weight, namely, 4° Cent. If we are finding 
the relative weight of a gas we use hydrogen gas as a 
standard. 

The specific gravity of a substance is therefore the 
ratio of the weight of a given volume of the substance 
to the weight of a similar volume of a substance taken 
as a standard. The specific gravity of a substance is 
therefore merely a number or a ratio and has no name 
attached to it. 

If W is the weight of a substance, S the weight of the 
same volume of a standard substance and g the specific 

W 

gravity of the substance we have the rule g = -^- From 

this rule W = g X S, that is, the weight of the sub- 
stance is equal to its specific gravity times the weight of 
the same volume of the standard substance. Also we can 

W 

write the equation in the form aS = — 

It is known that an object entirely immersed in a liquid 
loses in weight an amount equal to the weight of the 
volume of the liquid displaced by the object immersed. 
This principle enables us to determine easily the 
specific gravity of some substances by completely 
immersing them in liquids in which they are not soluble. 

Example, — As shown in Fig. 16, a small cast-iron 
cylinder weighs in air 1.12 lb. When completely 
immersed in water as shown its weight is only .965 lb. 
The loss in weight of the object due to submerging it in 



16 



MECHANICS AND ALLIED SUBJECTS 



water is therefore .155 lb. and its specific gravity equals 
weight in air 1.12 lb. 



loss of weight when immersed in water .155 lb. 



or 7.22. 



Evidently we could not use this method with water as 

a liquid if we wished to find 
the specific gravity of sugar, 
since sugar is soluble in water. 
In the case of substances 
which are dissolved by water 
we may use other liquids as 
the standard by which the 
substance is not dissolved and 
compare the weight of this 
new liquid with the weight 

of water in determining the specific gravity. 

Below is given a table of the approximate specific 

gravities of some of the most common materials. 

APPROXIMATE SPECIFIC GRAVITIES 

Iron, cast. ... 7.21 Copper 8 . 79 Granite 2.6 

Glass 2.9 

Pine, white . . .55 

Gold 19 . 26 Pine, yellow . . 66 



Fig. 16. 



Iron, wrought 7 . 78 Nickel 8.8 

Steel 7 . 92 Mercury, at 60° 13 . 6 



Lead 11.3 

Tin 7.29 Silver 10.47 Oak, white. . . .77 

Zinc 7.19 Aluminum... 2.67 Marble 2.7 

Brass 7.82' Cork 24 Ice 917 

Example. — Using the above table suppose we wish to 
find the weight of an iron casting containing 23^^ cu. ft. 

Solution. — One cubic foot of water weighs 62.5 lb. 
The specific gravity of cast iron is 7.21, that is, the cast 
iron is 7.21 times heavier than water, therefore the 
weight of the casting equals 



62.5 X 2.5 X 7,21 or 1126.56 lb. 



DENSITY AND SPECIFIC GRAVITY 17 

Example. — What is the weight of a white-oak beam 
4 in. X 8 in. and 18 ft. long? 

Solution. — We first find the volume of the beam in 

4 8 
cubic feet which is equal ^^ Jn ^ Tn X 18 or 4.0 cu. ft. 

Since from the table we see that the specific gravity 
of white oak is .77 that is, since it is .77 times as heavy 
as water the weight of the beam equals 

cubic feet X 62.5 X .77 

that is 4.0 X 62.5 X .77 or 192.5 lb. 

Example. — A timber is thrown into still water and it 
is found that it floats one-third out of water, what is its 
specific gravity? 

Solution. — Since the timber displaces a volume of 
water equal to % of its own volume it is held up or 
buoyed up by a force equal to the weight of a volume of 
water % that of the volume of the timber. The weight 
of the timber is therefore only % of the weight of the 
same volume of water, that is, its specific gravity is .67. 

Example. — What is the volume»of 100 lb. of copper? 

62.5 
Solution. — One cubic inch of water weighs TjK^ or 

.0361 lb. Since the specific gravity of copper is 8.79, 
1 cu. in. of copper weighs 8.79 X .0361 or .317 lb. 
There are therefore as many cubic inches of copper in 
100 lb. as .317 lb. the weight of 1 cu. in. of copper 

is contained into the total weight of copper or -^y^ = 315.4 

cu. in. 

PROBLEMS 

1. Using the table of specific gravities, find the weight of 8 
cu. ft. of wrought iron. 



18 MECHANICS AND ALLIED SUBJECTS 

2. Find the weight of a white pine timber 4 in. X 6 in. and 14 
ft. long. 

3. How^ many pounds heavier is a cubic foot of lead than a 
cubic foot of wrought iron? 

4. Find the number of cubic feet in a marble block w^eighing 
2^ tons. 

6. A cylindrical casting 23^ in. in diam. and 8 in. long weighs 
11.0 lb. What is its specific gravity? 

6. A tank car has a capacity of 50,000 lb. If it is to be filled 
with petroleum having a specific gravity of .832, how many cubic 
feet and how many gallons can the tank hold? 

7. The pressure of the atmosphere per square inch is about 
14.7 lb. How high a column of mercury 1 sq. in. in cross section 
is required to produce this pressure? 

8. A steel casting displaces 4.84 gal. of w^ater. What is its 
weight ? 

9. Two hundred cubic inches of brass are made up of 150 parts of 
copper and 50 parts of zinc by volume. Find its specific gravity. 
Solution: ^ 

The weight of 1 cu. in. of water = .0361 lb. 
The weight of 1 cu. in. of copper = 8.79 X .0361 or .317 lb. 
Therefore 150 cu. in. weigh 47.6 lb. The weight of 1 cu. in. of 
zinc = 7.19 X .0361 or .260 lb. Therefore 50 cu. in. weigh 
13.0 lb. 
If S is the specific gravitj' of the brass, 200 cu. in. weigh 

200 X .0361 X S lb. 

Since the weight of the brass equals the sum of the weights of 
its parts, we have, 200 X .0361 X S = 47.6 + 13 
therefore 

-4.7 (\ _L 1 "^ 

S = 200 X 0361 ^^ ^'^^ *^^ specific gravity. 

10. Forty pounds of copper and 27 lb. of zinc are alloyed to 
make brass. What is its specific gravity? 

11. A piece of metal weighs 26.8 lb. in air and 23.1 lb. when 
entirely immersed in water. What is its specific gravity? 

12. Find the weight of oil in a cylindrical tank 8 ft. in diam. 
and 16 ft. long if the specific gravity of the oil is .98. 



DENSITY AND SPECIFIC GRAVITY 19 

13. A casting weighs in air 22.8 lb. When entirely immersed 
in oil whose specific gravity is .94 the casting weighs 19.6 lb. 
Find its specific gravity and its volume in cubic inches. 

14. An alloy is made up of 30 cu. in. of tin and 48 cu. in. of 
copper. Find its specific gravity. 



CHAPTER IV 
SCREW THREADS 

10. Thread Sections. — The shape and size of a thread 
are determined by the shape and size of its cross section 
taken in a plane passing through the axis of the screw. 
The dimensions of any particular kind of thread are fixed 
bj^ stating the number of threads or sections in an inch, 
measured parallel to the axis of the screw, as, for example, 
8 threads per inch, or simply 8 threads. 

Fig. 17 shows the sections of four different screw 
threads. In the machine shop on regular work the U. S. 




V. V////////////A ///////////// 
V' THREAO U, 5. STP, S(^UARE 

Fig. 17. 

Std. Thread is used. In some practice bolts and studs 
Y^ in. in diam. and above screwed into boilers have 
twelve sharp V threads per in., irrespective of the diam- 
eter of the bolt or stud. 

11. Pitch and Lead. — Before looking into the method 
of constructing and calculating threads we should under- 
stand first the meaning of the ^^ pitch'' and '^lead'' of 
a screw. The ^^ pitch'' of a screw is the distance from 
the center of one thread to the center of the next. The 
^^lead" of a screw is the distance the screw advances in 
one complete turn. 

20 



SCREW THREADS 



21 



12. Single, Double, and Triple Threads. — A single- 
threaded screw, as shown in Fig. 18, is one which ad- 
vances in one complete turn a distance (measured in the 
direction of the axis of the screw) equal to that from a 
point on one thread to a corresponding point on the next 
thread, or a distance equal to the ^^ pitch/' 

A double-threaded screw, as shown in Fig. 19, ad- 




S/^Oi£ THREAD 
Fig. 18. 




DOUBLE THREAD 

Fig. 19. 




TRIPLE THREAD 
Fig. 20. 




I I ! 

QUADRUPLE THREAD 
Fig. 21. 



vances in one complete turn a distance equal to twice the 
^^ pitch.'' For a double thread the '4ead" is therefore 
twice the ^^ pitch." In a triple thread, as shown in Fig. 
20, the advance for one complete turn equals three times 
the '^ pitch," or the ^' lead " equals three times the '^ pitch." 
Fig. 21 shows a quadruple thread. 

13. Right- and Left-handed Threads. — A right-handed 
thread is one in which the thread rises from left to right. 



22 



MECHANICS AND ALLIED SUBJECTS 



as from A to B in Fig. 22. Fig. 23 sliows a left-handed 

thread. 

Suppose now, for example, we were making a sketch 

of a threaded bolt, we should show the following. The 

total length of bolt including the head, the length of 

body including the threaded por- 
tion, length of threaded portion, 
diameter of bolt, the thread sec- 
tion, number of threads per in., 
whether thread is right- or left- 
handed and whether single, dou- 
ble or triple. If there are no 
standard tables available for find- 
ing the dimensions of the bolt 
and short diameter of the head 



z' V^ - S^ -s 





Figs. 22 and 23. 



head, the thickness 
should be shown. 

14. Construction of a Single, Right-handed V Thread, 
8 Threads to an Inch. — ^Lay off on line ABj Fig. 24 (an 
element of the portion to be threaded), 3^^-in. distances as 
1-2, 2-3, etc., each equal to 
the pitch of the screw. With 
the 60° triangle draw the sec- 
tions la2, 263, etc. A line 
through point a parallel to line 
AB contains the points b, c, 
d, etc., at the bottoms of the 
threads. A perpendicular 

from point a to line CD gives point 5 on the outside of 
the thread. Similarly perpendiculars dropped from 
points by c, d, etc., locate points on the outside of the 
thread on line CD. With the 60° triangle the sections 
5^6, etc., are completed and the visible edges of the thread 
are drawn by connecting points b and e , 2 and 5, etc., as 





SCREW THREADS 23 

shown. The edges of the tliread form turns called 
^^ helices/' a single turn being a ^^ helix/' These edges 
are, however,, simply represented by straight lines which 
are used in most cases as in Fig. 24. 

15. Construction of a Double, Left-handed V Thread, 
4 Threads to an Inch. — In Fig. 25 portion A 5 is to 
be threaded. The ''pitch'' is here }y^ in., and the 
'^lead'^ H in. Since the thread is double, the outside 
points of the thread on opposite sides of the screw fall 
opposite each other. The construction is otherwise the 
same as for the single V thread. 

A F R 

In one complete turn the screw 

advances a distance equal to 

twice the ''pitch," or a distance 

of Yi in. equal to the "lead" 

of the screw. The distance ^E ^^^^ o- 

corresponds to the pitch. The 

screw is left-handed, since the thread ascends from right 

to left looking toward the head of the bolt. 

16. Construction of a United States Standard Thread. 
— The U. S. Std. for screws and threads is that adopted 
by the U. S. Government and by the R. R. Master 
Mechanics and Master Car Builders Associations, also 
by leading tool builders and manufacturers. This thread 
has the same angle as the V thread or 60° thread. The 
dimensions of a thread for a pitch equal to 1 in. are 
shown in Fig. 26. The triangle ABC is equilateral. If 
each of its sides equals 1 in. the distance CD equals the 
square root of [(1)^ — (J^^)^] or the square root of (1 — 
0.25) or 0.866 in. This is therefore the depth of a V 
thread of pitch equal to one. In this U. S. Std. thread 
the depth EF = 0.65 in., that is, for a pitch of 1 in. 
The distance across the flats at top and bottom is 0.125 



24 



MECHANICS AND ALLIED SUBJECTS 



in., as shown in the figure, that is, J^ of the pitch. To 
find the depth of a U. S. Std. thread of another pitch, 
divide these figures by the number of threads to the inch. 



j^AYc/^V' 




Fig. 26. 



PROBLEMS 



1. What is the lead of a single-threaded screw having 12 threads 
per in.? What is the pitch of this screw? 

2. A double thread measures .125 in. between the points, 
what is its pitch and lead? 

3. How many turns must a single thread of pitch .166 in. make 
to advance l}i in.? 

4. A double-threaded screw advances IJ2 in. in three complete 
turns. What is its pitch? 

5. Find the depth of thread, and width of flats for a U. S. Std. 
Thread, with the following number of threads per in. 8, 12, 13, 18. 

6. The number of threads per inch (U. S. Std.) corresponding 
to the diameter of the bolt are as follows: — Diam. of bolt J^ in.; 
threads per in., 13; diam.%in.; threads, 9; 13-^ in. diam. threads, 7. 
For each of these cases calculate the diameter at root of thread 
and the area of bolt at root of thread, also the width of flat. 

7. In a machine, three bolts each 1 in. in diam. with 8 U. S. 
Std. threads per in., are replaced by a single bolt. What must be 
the size of this bolt at root of thread to have the same area of 
metal at root of thread as the three 1-in. bolts? 

8. What is the ^'pitch" and the ''lead^' of a triple thread which 
advances 2 in. in two complete turns? 

9. Make a sketch of a double, right-hand V thread with a 
pitch of }i in. 



SCREW THREADS 25 

10. Find the size of a tap drill in nearest 64ths in. for a ^' V" 
thread 8 per in. on a bolt 1 in. in diam. 

11. Make a sketch of a single right-hand U. S. Std. thread with 
8 threads per in. What is the pitch and the lead in this case? 

12. How does the number of threads per in. on a bolt and hence 
the depth of the thread affect the strength of the bolt? Would 
it be practicable to have only 4 threads per in. on a 1-in. bolt? 
Why with some companies are. 12 sharp V threads per in. used in 
boiler work on any size bolt or stud above ^i in. in diam.? 

13. What is the advantage of double and triple threads over a 
single thread? How does the number of threads per in. affect 
the '^ speed" of a bolt or screw? Which form of thread is used 
most in the machine-shop work, the V or U. S. Std.? 

Referring to Fig. 27, showing the section of a V thread, 
the distance AB, or the diameter at the root of the thread 
equals the diameter of ^^ ,^ 

the bolt minus twice the r-nr 
distanceCD. The length / ^' 
CD is .86 of the distance 1 '^ 
EF, and the distance EF l\ | 
depends on the number I J "^ 
of threads per in. of the 
screw. In an 8-thread 
screw EF = }^ in. or .125 in., for a 10-t bread screw EF = 

}io in. or .10 in., for a T-thread screw EF = ^ 

From the above we have the rule for a V thread. 

where R = Diam. at root of thread. 

D = Diam, of bolt (outside of threads). 
T = No. of teeth or threads per in. 

If in the case of bolts or studs 'j^^ in. in diam. and above, 
screwed into boilers, 12 sharp V threads per in. are used, 



Fig. 27. 



26 



MECHANICS AND ALLIED SUBJECTS 



irrespective of the diameter of the bolt or stud, the pre- 
ceding rule becomes in this case: 

R=D-^ or R = D 



12 



0.144 m. 
or R = D - %4^ in. 



In the case of the U. S. Std. thread as in Fig. 28 the 
diameter at the root of the thread equals the diameter 
of the bolt minus twice the distance AB. The distance 
AB = .65 X the pitch. Therefore the diameter at the 
root of the thread 



Pi-hcb ,of ihreacf 




Fig. 28. 

= diameter of bolt — 2 X 0.65 X the pitch. 

The pitch = rf, where T is the number of threads 

per ino 

Therefore diameter at root of thread for U, S. Std. 

= diameter of bolt 7^7- 



17. Construction of a Single, Square Thread, 2 Threads 
Per Inch. Fig. 29 Shows the Method of Construction. 

18. Construction of an Acme Thread. — Square threads 
are not always easily cut, and there has been developed a 
standard, which is flat at top and bottom and has its 
sides inclined at an angle of 14)^° to the center line of 



SCREW THREADS 



27 



the thread section. Fig. 30 shows the section for such 
a thread and the dimensions of the threads for a pitch 
of 1 in. 

The angle of 29° or the total angle formed by the sides 



\<-p/fch ^: 




Fig. 29. 



of an Acme thread as in Fig. 31 may be drawn with the 
use of a protractor or as shown in the figure which 
is drawn as follows: 







-0.366" 
Fig. 30. 




A circle ABCD is drawn 2 in. in diam. and also a vertical 
diam. AC, From A as a center and with a radius of 
3^^ in., arcs are drawn cutting the circle at points e and /. 
Connecting these points with point C by the straight 



28 



MECHANICS AND ALLIED SUBJECTS 



lines eC and fC gives the inclination for the sides of the 
threads. The angle eCf is 29°. 

In a ^^Worm^' thread as in Fig. 32 the angle is the 
same as for the Acme thread, that is, 29°, but the thread 
is about one-third (}i) deeper than the Acme. 



29 



n .0.335 




r<r 0.69^0.31^ 

Fig. 32. 



k-/^-H 



19. Miscellaneous Rules. — The width across flats of 
U. S. Std. hex. nuts equals 13-^ X Diameter of bolt + 3^^ 
in. See Fig. 33. The width across corners of U. S. 
Std. hex. nuts equals 1.155 X width across flats. The 



\^0.8I^^ 




T 




^ 
^ 



J..-\ 



Fig. 33. 



thickness of U. S. Std. hex. nuts (rough) is the same as 
diameter of bolt. Width across corners and flats of bolt 
heads is the same as for hex. nuts. The thickness of 
bolt heads equals half the width across flats of bolt head 
or % X diameter of bolt + 3^f e in. 



SCREW THREADS 



29 



Table for U.S. Std. Threads 



Diam. 
of screw 


Threads 
to inch 


Pitch 


Depth of 
thread 


Diam. at root 
of thread 


Width of 
fiat . 


H 


20 


.0500 


.0325 


.185 


.0063 


He 


18 


.0556 


.0361 


.2403 


.0069 


% 


16 


.0625 


.0405 


.2936 


.0078 


Ke 


14 


.0714 


.0461 


.3447 


.0089 


V2 


13 


.0769 


.0499 


.4001 


.0096 


He 


12 


.0833 


.0541 


.4542 


.0104 


% 


11 


.0909 


.0591 


.5069 


.0114 


M 


10 


.1000 


.0649 


.6201 


.0125 


Vs 


9 


.1111 


.0721 


.7307 


.0139 


1 


8 


.1250 


.0812 


.8376 


.0156 


IH 


7 


.1429 


.0928 


.9394 


.0179 


IK 


7 


.1429 


.0928 


1.0644 


.0179 


i^g 


6 


.1667 


.1082 


1 . 1585 


.0208 


i>^ 


6 


.1667 


.1082 


1.2835 


.0208 


1% 


5M 


.1818 


.1181 


1 . 3888 


.0227 


IM 


5 


.2000 


.1299 


1.4902 


.0250 


1% 


5 


.2000 


.1299 


1.6152 


.0250 


2 


4M 


.2222 


.1444 


1.7113 


.0278 


2M 


41^ 


.2222 


.1444 


1.9613 


.0278 


2K 


4 


.2500 


.1624 


2.1752 


.0313 


2?i 


4 


.2500 


.1624 


2.4252 


.0313 


3 


3M 


.2857 


.1856 


. 2.6288 


.0357 


3M 


3>^ 


.2857 


.1856 


2.8788 


.0357 


33-^ 


33-i 


.3077 


.1998 


3.1003 


.0385 


Wi 


3 


.3333 


.2165 


3.3170 


.0417 


4 


3 


.3333 


.2165 


3.5670 


.0417 


4Ji 


2% 


.3478 


.2259 


3.7982 


.0435 


4>^ 


2% 


.3636 


.2362 


4.0276 


.0455 


4% 


2?^ 


.3810 


.2474 


4.2551 


.0476 


5 


2>^ 


.4000 


.2598 


4.4804 


.0500 


5M 


2M 


.4000 


.2598 


4.7304 


.0500 


5>^ 


2^^ 


.4210 


.2735 


4.9530 


.0526 


5^i 


2% 


.4210 


.2735 


5.2030 


.0526 


6 


2H 


.4444 


.2882 


5.4226 


.0556 



30 



MECHANICS AND ALLIED SUBJECTS 





Tabld for Sharp V 


Threads 




Diam. 
of screw 


1 No. threads 
per inch 


j Pitch 

1 


Depth of 
thread 


Diam. at root 
of thread 


Vi 


20 


.0500 


.0433 


.1634 


He 


18 


.0556 


.0481 


.2163 


H 


16 


.0625 


.0541 


.2667 


Ke 


14 


.0714 


.0618 


.3140 


V2 


12 


.0833 


.0722 


.3557 


%6 


12 


.0833 


.0722 


.4182 


% 


11 


.0909 


.0787 


.4676 


iHe 


11 


.0909 


.0787 


.5301 


Vi 


10 


.1000 


.0866 


.5768 


^He 


10 


.1000 


.0866 


.6393 


% 


9 


.1111 


.0962 


.6826 


^He 


9 


.1111 


.0962 


.7451 


1 


8 


.1250 


.1083 


.7835 


IM 


7 


.1429 


.1237 


.8776 


m 


7 


. 1429 


" .1237 


1 . 0026 


\% 


6 


.1667 


.1443 


1.0864 


W2 


6 


.1667 


.1443 


1.2114 


\% 


5 


.2000 


.1733 


1.2784 


m 


5 


.2000 


.1733 


1.4034 


V/s 


4K 


.2222 


.1924 


1 . 4902 


2 


4>^ 


.2222 


.1924 


1.6152 


2>^ 


4>^ 


.2222 


.1924 


1 . 7402 


2^ 


4>^ 


.2222 


.1924 


1.8652 


2% 


4>^ 


.2222 


. 1924 


1.9902 


2>^ 


4 


.2500 


.2165 


2.0670 


2H 


4 


.2500 


.2165 


2.1920 


2y^ 


4 


.2500 


.2165 


2.3170 


2% 


4 


.2500 


.2165 


2.4420 


33 


3>^ 


.2857 


.2474 


2.5052 


% 


3>^ 


.2857 


.2474 


2.6301 


SH 


3K 


.2857 


.2474 


2.7551 


3% 


3>i 


.3077 


.2666 


2.8418 


3H 


3K 


.3077 


.2666 


2.9668 


3H 


3>i 


.3077 


.2666 


3.0918 


3H 


3 


.3333 


.2886 


3.1727 


SVs 


3 


.3333 


.2886 


3.2977 


4 


3 


.3333 


.2886 


3.4227 



CHAPTER V 

CALCULATION OF LEVERS 

20. Definitions and Problems. — A lever is a rod, bar or 
beam of any shape, either straight or curved, which is 
free to turn about a fixed point called a ^^ fulcrum/' The 
fulcrum may be any form of support^ prop or bearing. 




V 



Fig. 34. 



Levers are divided into three classes according to the 
positions of the loads, pressures, or forces acting on the 
lever with respect to the position of the fulcrum. In 



iA 




I 



Fig. 35. 



the first class, Fig. 34 the fulcrum A is between the 
weight (W) and the force (F) applied to lift the weight. 
In the second-class lever Fig. 35 the weight W is 
between the fulcrum A and the force F, 

31 



32 



MECHANICS AND ALLIED SUBJECTS 



In the third-class lever the force F is between the ful- 
crum A and the weight W, as in Fig. 36. 

In working out lever problems we find what are called 
the ^^ moments ^^ of the forces or pressures or weights 
acting on the lever. In Fig. 37 which shows a lever of 
the first class the force F, 4 ft. from the fulcrum A, 



^ 




W 



Fig. 36. 



raises the weight W of 100 lb. located 2 ft. from A. 
Multiplying i^ by 4 lb., its distance from A gives the 
moment of F about the ^^fulcrum'^ A. Similarly TF X 2 
ft. is the moment of W about the ^^ fulcrum ^^ A. To 
find what force is necessary at F to raise a weight of 
100 lt>. at W, we multiply i^ X 4 ft. giving the moment of 



■>K- 



■^ 



A 



r 



7S" 
A 



5 



s 



■>', 



Fig. 37. 



W=500 

Fig. 38. 



F about A and place this product equal to 100 X 2 
which is the moment of the weight of 100 lb. about A. 
Then 

100 X 2 = 7^ X 4 

therefore 200 = 4F 



and 



7^ = 2-?^ or 50 lb. 



CALCULATION OF LEVERS 33 

Therefore 50 lb. pull at F will lift 100 lb. at W neglect- 
ing friction and the weight of the lever itself. The rule 
in any case is, find the moment of the force about the 
fulcrum and place this f^^soo 

equal to the moment of ^^ i^ y y[ 




- 9' 



i 



Y/^/00 



the weight about the 
fulcrum and solve for 
the quantity desired. Ym 39 

In Fig. 38 to find a 
force F necessary to lift a weight of 500 lb. at the dis- 
tance given we have 500 X 2 = /^ X 8 therefore 

1000 = ^F 
and F = 125 lb. 

In Fig. 39 which represents a lever of the third 
class a force of 300 lb. at F will lift only 100 lb. at W, 
since taking the ^^ moments^' about the fulcrum A we 
have 300 X 3 = Tf X 9. 

Therefore 900 = OTT 

and W = 100 lb. 

PROBLEMS 

What class of lever does Fig. 40 show and why? From the 
rules already given and using Fig. 40, fill out the following table, 



< a ?^ 



^ , I 



w 

Fig. 40. 

giving also sample calculations to show how you obtained your 
results. Neglect the weight of the lever itself. 
3 



34 



MECHANICS AND ALLIED SUBJECTS 



No. 


IF, lb. 


F, lb. 

1 


a 

i 


b 


1 


30 




1 ft. 


3 ft. 


2 




50 


2 ft. 


8 ft. 


3 


60 


30 




6 ft. 


4 


80 


10 


2 ft. 




5 


18 




33^ ft. 


m ft. 


6 




40 


Sin. 


2 ft. 



What class of lever does Fig. 41 show and why? Using Fig. 
41 fill out the following table, giving also sample calculations to 



t 



K- 



a 



->K- 






? 



Fig. 41. 



show how you obtained 3^our results. Neglect the weight of the 
lever itself. 



No. 


i^lb. 


W\h. 


a 


b 


c 


7 


50 


200 


8 ft. 






8 


30 




6 ft. 




2 ft. 


9 




10 


1ft. 


5 ft. 




10 


30 


60 




4ft. 




11 




20 


6^ ft. 




2 ft. 


12 


27 






39 in. 


27 in. 



What class of lever does Fig. 42 show, and why? Using Fig. 
42 fill out the following table, giving also sample calculations to 
show how you obtained your results. Neglect the weight of the 
lever itself. 



CALCULATION OF LEVERS 



35 



y. 



->i 



Z] 



Fig. 42. 



W 



No. 


W, lb. 


F, lb. 


a 


b 


c 


13 


50 


200 


8 ft. 






14 


30 




Oft. 




2 ft. 


15 




10 


10 ft. 


5 ft. 




16 


30 


60 




4 ft. 




17 




20 


QH ft. 




2 ft. 


18 


27 




• 


39 in. 


27 in. 



19. As shown in Fig. 43, what downward force at F is necessary 
to start the stone if the resistance offered is 300 lb. If the stone 




could not be moved with the bar as shown how could a greater 
force or pressure be produced on the stone, using the same bar? 
What class of lever does this represent? 

20. In Fig. 44 the lever arms are bent at right angles to each 
other, allowing the effect of the weight W to be produced in a 
horizontal direction at F. With the lever arms as shown, what 
weight is necessary at W to produce a pull of 100 lb. at F? 

21. In Fig. 45, showing a safety valve for a stationary boiler, 



36 



MECHANICS AND ALLIED SUBJECTS 



at what pressure will the boiler ^'blow off" with the size of valve 
and lever arms as shown? What class of lever does this show? 









\ 


1 


^ 




.^^ .^'^ 


J 






/u 


1 




w=? 













Fig. 44. 







21. Calculation for Levers Taking into Account the 
Weight of the Lever Itself in Each Case. — If the lever 
shown in Fig. 46 weighs .5 lb. per in. of length the por- 
tion to the left of the fulcrum weighs 8 X .5 or 4 lb. 



8' 



->K- 



Fulcrum 



24 



^ 



■zpr 



-4-^ 



It 



trn 



ll'Ib. 



F-? 



tSO-lb. 



Fig. 46. 



The portion to the right of the fulcrum weighs 24 X .5 
or 12 lb. These weights should be taken at the center of 
gravity of the respective parts of the lever. 

For example, the entire weight of the 8-in. length is 



CALCULATION OF LEVERS 



37 



considered as concentrated at a point 4 in. from the fulcrum, 
that is, at the center of gravity of the 8-in. length. Like- 
wise the 12 lb. is considered as concentrated at a distance 
of 12 in. from and at the right of the fulcrum. We now 
have four forces to deal with and our equation becomes: 



1^ 








J^" 


F'? 

• — >- 


1 .Fulcrum 


T' -7" 








!i ^ 12' 


f 








^& 








12-lb. 























200-7t>. 

Fig. 47, 



150 X 8 + 4 X 4 

1200 + 16 

1200 + 16 - 144 

1072 

44.7 lb. 



12 X 12+F X24 

144 + 24F 

24F 

24F 

F 



Fulcrum 



^ 



s 



r 



12 



£ 



12-lb 



2-? 



200-lb. 



Fig. 48. 



In Fig. 47, the lever weighs .5 lb. per in. of length or 
12 lb. total weight. This acts at a distance of 12 in. 
from the fulcrum, that is, at the center of gravity of the 
whole lever. The equation then becomes 

200 X 8 + 12 X 12 = F X 24 
1744 = 24F 
72.6 lb. = F 



38 



MECHANICS AND ALLIED SUBJECTS 



In Fig. 48, the lever weighs 12 lb. and the equation 
becomes 

12 X 12 + 200 X 24 = 8F 

4944 = SF 

6181b. = F 



PROBLEMS 

22. In a lever of the first class a weight of 250 lb. 6 in. from 
the fulcrum can l^e balanced by what force applied IJ^ ft. from 
the fulcrum? 

23. Work out the above problem if the lever itself weighs 4 lb. 
per ft. of length. 




24. Find the force F required to lift the weight of 200 lb. on 
the wheelbarrow as shown in Fig. 49. 

26. From Fig. 50 find the force which can be exerted at F through 
the air cylinder and system of levers shown. 



,i 



12" D I am. 
60-lb.persq.m, 



CO 



Fulcrum 



Fig. 50. 



^1. 



— >• 



^- -^fulcrum 



26. From Fig. 51, showing a safety valve for a stationary boiler, 
what must be the length of the arm ''A" in order that the boiler 
will ''blow off" at 150 lb. pressure per sq. in.? What class of 
lever does this sketch show? 



CALCULATION OF LEVERS 



39 



27. From Fig. 52, showing a safety valve for a stationary boiler, 
what weight must be placed at W in order that the boiler will 
''blow off" at 150 lb. pressure per sq. in.? 



^ 



SOib. 




.^^^f^^^a 



Fig. 51. 



^ 



W'? 



18" 



M 



fc 






2''D/am 



Fig. 52. 




28. In using an ordinary claw hammer for pulling a nail as shown 
in Fig. 53, we are using a first-class lever. Suppose as shown 



40 MECHANICS AND ALLIED SUBJECTS 




rum 



Fig. 54. 




Fig 55. 



CALCULATION OF LEVERS 



41 



a force of 40 lb. is exerted at ^^F/' we find the resistance of the 
nail at the start by calculating a first-class lever, whose arms are 
bent up at right angles to each other. If the distance s = 2 in. 
and S = 9 in., calculate the resistance " R'' of the nail at the 
start. 

29. The pliers and wire cutters shown in Fig. 54 form a first-class 
lever. If the total force on the handle of the pliers is 10 lb. at a 
distance of 33^^ in. from the fulcrum, how much pressure is produced 
on the wire at "P'^ in cutting it? The wire is % in. from the 
fulcrum. 

30. Fig. 55 shows a ^'belt shifter.'' If the resistance at ''R,'' 
required to be overcome in shifting the belt is 35 lb., what force 
is required at ^^/^" to move the belt with the lever arms shown? 



Fu/cruw I 



'-^ 




31. If in shifting the belt it is necessary to move the shifter rod 
4 in. how far does the end of the long arm move, that is, the end on 
which the force ''/^" is exerted? 



In Fig. 56 is shown a second-class brake lever with 
brake shoe and break rod. The motion of the brake 
shoe is to the motion of the pull rod as 18:24 with the 
lengths of lever arms as shown. For example if the 



P 24 
called '^P^^ we have the proportion tt = t^ from which 



42 MECHANICS AND ALLIED SUBJECTS 

brake rod moves 4 in. the shoe movement is found as 
follows: 

Movement of brake shoe 18 movement of brake shoe 

_ — ;= — QY — — — — — — — 

Movement of pull rod 24 4 

= 18 

~ 24 

18 
That is, movement of brake shoe = 4 X ^ = 3 in. 

Also if the braking force through the brake shoe is 

F_X24 
18 
Example. — If the pull on the rod is 4800 lb., that is, 

4-ROO V 24 
if F = 4800 lb. then P = ^^^ or 6400 lb. 

32. In the brake lever shown in Fig. 56, if the distance from the 
fulcrum to brake shoe is 24 in. and the whole effective length of 
lever is 30 in. what is the movement of brake shoe compared with 
that of the brake rod? What pull is necessary to produce a pres- 
sure of 8000 lb. against the wheel? 

33. Find the length of lever arms for a brake lever as shown in 
Fig. 56 such that a pull of 800 lb. on the brake rod will produce a 
pressure of 4000 lb. of the brake shoe against the wheel. In this 
case what is the ratio of movement of brake shoe and brake rod? 

Levers used to Increase or Decrease Motion, — Levers are 
often used to increase motion which of itself is too small 
to read easily as in the case of an indicator as shown in 
Fig. 57. With this instrument the piece to be measured 
is put in at ^'GJ^ The instrument is ^^ calibrated'^ or 
compared with a standard gauge to find the reading on 
the scale '^S^^ for a given gauge. When the piece is put in 
a ^'G^^ any movement of the pointer above or below the 
gauge point on ^^>S'' shows how much too large or too 



CALCULATION OF LEVERS 



43 



small the piece is that is being measured. The reading 
of the scale for different pieces tested will depend on 
the ratio of the arms of the lever used. If the short arm 
is .5 in. and the long arm 10 in. as shown the ratio of 
the motion of the ends of the lever is as 1 : 20. If for 



•^ik 




Fig. 57. 

example in testing a piece the end of the long lever moves 
.2 in. beyond the standard gauge mark, the work being 
measured is one-twentieth of .2 or .01 in. from the 
standard. 

PROBLEMS 

34. In the indicator shown in Fig. 57 the pointer indicates on 
scale 'VS" a distance of .48 in. beyond the gauge mark. How 
much is the piece being measured out of gauge? 




^/^\ 



Fig. 58. 



36. An indicator similar to that shown in Fig. 57 has .lever arms 
of .75 in. and 18 in. If a piece being measured is .025 in. from 
standard gauge, how far will the pointer move on scale " S^^ from 
the standard gauge mark? 



44 



MECHANICS AND ALLIED SUBJECTS 








t*-/0^ 



W'- 
lOOO-lb. 



Fig. 59. 



Fig. 60. 



1-t 



Fig. 61. 



F^IOO-Ib] 



F^ao-Jb, 



^ 



10-^ 



I/O 



100 



Fig. 62. 



-A 




Fig. 63. 



f"-60-/b 



CALCULATION OF LEVERS 



45 



36. In the lever indicator system shown in Fig. bS, find the mo- 
tion produced at '' S'' by a motion of .040 in. at G. 

37. Show how you could redesign the levers so that a motion 
of .040 in. at G would produce at ''S'' twice the motion calculated 
in the problem above. 








€^ 



F^ 



-=^ 



-<s> 



W-300-/1?, 



Fig. 65. 



38. In Fig. 59, find the force F necessary to raise the weight W, 

39. In Fig. 60, find the length of the lever arm X so that the 
force of 100 lb. will balance the load W of 1500 lb. 

40. In Fig, 61, find the weight W that can be raised by the force 
of 80 lb. 



46 



MECHANICS AND ALLIED SUBJECTS 

J2 'Brake C^lmder 
/ '' ^O'lb. Ptessure perscf. In. 



Find /he force 
produced on fhe 
^ brake rod ai^'W" 




Fig. 66. 




^"--^ rroniliff 
sha-Ff^ 
bearing 



'Radius 
Rod 



/" ^j^-' reverse spnng 

T 

Fig. 67. 



CALCULATION OF LEVERS 47 

41. In Fig. G2, find the force F required to raise the weight W. 

42. In Fig. 63, showing a pinch bar 6 ft. long with 2-in. nose, 
find the weight W that can be raised. 

43. In Fig. 64, showing a throttle lever, find the force F required 
to move the throttle. 

44. Fig. 65 shows a reverse lever. Find the force F required to 
reverse the engine. 

46. Fig. 66 shows a brake cylinder. Find the force produced 
on the brake rod at W by the pressure of 50 lb. per sq. in. in the 
cylinder. 

46. With a load at W, Fig. 67, of 400 lb. due to lifting the link 
blocks, radius rod ends and moving the valves, what will be the 
force required at P to move the reverse lever backward taking into 
account the action of the reverse spring, but neglecting the friction 
of the reversing gear? 



CHAPTER VI 



PULLEYS (BLOCK AND TACKLE) 



22. Arrangement of Pulley Systems. — Pulleys or 
blocks (Fig. 68) are used to change the direction of appli- 
cation of a force, as well as to reduce the value of the force 
necessary to lift a given weight or overcome a given 
resistance. Pulleys are in fact rotating levers. 

With a single pulley as in Fig. 68, 
the force F applied to raise the weight ///////////<;/// 
W, can be exerted downward instead 
of upward. This makes it easier to 



////////////, 



/////////////y 






raise the weight. In this case the force required at F is 
equal to the weight lifted at TF, neglecting friction. 

This arrangement forms a lever of the first class. Com- 
pare this with a first-class lever. 

In the case of a single movable pulley, as in Fig. 69, 
the force necessary at F is one-half the weight lifted at 

48 



PULLEYS {BLOCK AND TACKLE) 



49 



Wy neglecting the effect of friction. This arrange- 
ment forms a lever of the second class and we have the 
weight W X radius of pulley = F X diameter of 
pulley. Therefore to lift 100 lb. at W requires a force 
of 50 lb. at F. Compare this figure with a second-class 
lever. 

Fig. 70 shows a fixed and a movable pulley, or a 
combination of the arrangements shown in Figs. 68 
and 69. The force required at F to lift a weight W is 



////////7/// 




///// / /////A 



F^ISO 




///////////// 



F'-iei ^ 




F-'/2S 



Fig. 71. 



Fig. 72. 



W--500 
Fig. 73. 



equal to one-half of the weight neglecting friction of 
the rope and pulleys. When the force F is exerted 
through a distance of 2 ft., the weight is lifted 1 ft. A 
force of 50 lb. at F will raise a weight of 100 lb. at TF. 
The friction of the rope on the pulleys and the pulleys 
on their bearings is usually small compared to the 
weights ordinarily raised, and this friction is reduced by 
the use of grease or oil on the bearings. 

In the following arrangements of blocks. Figs. 71, 
72, and 73, the force required to raise a given weight 

4 



50 MECHANICS AND ALLIED SUBJECTS 

or vice versa is found as follows: Rule, — The force in 
pounds multiplied by the distance in feet through 
which it moves, equals the weight in pounds lifted, multi- 
plied by the distance through which it moves, and 
another rule which also applies for any number of 
pulleys in either block is: Divide the weight to be lifted 
by the number of single strands or lines of rope ex- 
tending down to the movable pulley, to find the force F, 
The distance through which the weight moves equals 
the distance through which the force F moves divided 
by the number of single lines of rope extending down to 
the movable block. In Fig. 71 a force of 250 lb. is 
necessary to lift a weight of 500 lb., neglecting friction. 
For each of the arrangements shown the force nec- 
essary to raise a weight of 500 lb. is given. 

PROBLEMS 

1. With a single movable pulley as in Fig. 69, what force is 
necessary at F to raise a weight of 300 lb.? What advantage has 
the arrangement shown in Fig. 70 over this one using only the 
single movable pulley? 

2. In Fig. 71 what force at F is necessary to lift 100 lb. at TF? 
What weight at W will just balance a downward pull at F of 230 
lb.? 

3. With arrangement as shown in Figs. 72 and 73, does the 
size of pulleys have anything to do with the force required to lift 
a given weight? 

4. In Fig. 72, how much force is necessary at F to raise a weight 
of 800 lb. at W? 

6. In Fig. 72, what weight at W will just balance a pull at F 
of 200 lb. if the lower block and hook together weigh 20 lb.? 

6. In Fig. 73, what weight could a man weighing 150 lb. 
raise by hanging on the rope at the end F? 

If the weight was raised 1 ft., through how many feet does the 
man move? 

7. What force at F will a weight of 1000 lb. at W just balance 



PULLEYS {BLOCK AND TACKLE) 



51 



with the arrangement of Fig. 73, if the lower block and hook 
together weigh 26 lb.? 

The Differential Pulley, — For lifting heavy weights by 
hand a differential pulley is often used. 

As shown in Fig. 74, this consists of two pulleys A 
and B rigidly fastened together and rotating as one 
about a fixed axis C, An endless chain passes over both 
pulleys, and one of the loops of the chain passes under 
and supports the lower or movable block D, The other 
loop of the chain hangs freely and is the loop upon which 
one pulls in raising a weight on the hook 
E, The rims of the pulleys are grooved 
and carry lugs to prevent the chain from 
slipping. If the radii of the large and 
small fixed pulleys are R and r respec- 
tively, ^^P^^ the pulling force exerted on 
the chain and ^'W^^ the weight lifted, 
the rule for the pulley is as follows, leav- 
ing out friction: 



or 



PXR = W X VziR 
W ^ R 

P - mR - r) 



r) 




As used a differential pulley can ac- 
tually lift only about 25 to 35 per cent, 
of the weight calculated for a given pull 
from the formula stated above. This means therefore 
that to lift a given weight requires actually from three 
to four times the pull calculated by the formula which 
does not take account of friction. 

Example, — What force is necessary to lift a weight of 
800 lb. with a differential hoist having pulleys 10 in. 
and S}^ in. in diam. and an efficiency of 25 per cent.? 



52 MECHANICS AND ALLIED SUBJECTS 

Solution.— P X R = W X HiR - r) 

W XViiR- r) 



that is P = 
orP = 



R 

800 X 3-^(5 - 41^) 

5 
800 XV2XV4, 



800 X % _ ,, 
or z or 60 lb. 



Since the efficiency is only 25 per cent, the actual pull 
required is -^ or 240 lb. 



PROBLEMS 

8. From the rule for a differential pulley find the force necessary 
to lift a tank weighing 280 lb. The diameters of the pulleys being 
12 in. and 10 in. and the efficiency 30 per cent. 

9. In a differential hoist with pulleys 11 in. and 93^^ in. and an 
efficiency of 53 per cent., find what weight a 150-lb. man can raise 
by putting his own weight on the chain. 



CHAPTER VII 

THE INCLINED PLANE AND WEDGE, 
SCREW JACK 



THE 



23. The Inclined Plane and Wedge. — A block whose 
sloping surface is less than 90° may be considered an ^^ in- 
clined plane.'' In Fig. 75 which shows such an inclined 
plane, AB is the ^^base/' BC the ^^ height'' and AC the 
^^face" or ^^ sloping surface." With the use of the 
inclined plane a given resistance can be overcome with 
a smaller force than if the plane were not used. For ex- 





FiG. 76. 



ample, in Fig. 76, suppose we wished to raise a w^eight 
of 1000 lb. through the vertical distance BC = 2 ft. 
If this weight were raised vertically and without the 
use of the plane the force of 1000 lb. would have to be 
exerted through the distance BC, If, however, the 
inclined plane is used and the weight is moved over its 
face AC, sl force of only % of 1000 lb. or 667 lb. is nec- 
essary, although this force is exerted through a dis- 
tance AC which is greater than distance BC. With the 

53 



54 MECHANICS AND ALLIED SUBJECTS 

inclined plane, we, therefore, require a smaller force 
which must be exerted through a greater distance to do a 
certain amount of work. 

Letting F represent the force required to raise a given 
weight on the inclined plane, and W the weight to be 
raised, we have the proportion: 

F height of plane 

W length of sloping surface of plane 

or for Fig. 76 

F^ 2 J^ _ 2 

Tf ~ 3 1000 " 3 

therefore F = 2^ X 1000 or 667 lb. 

Stated in words the rule for the '^ Inclined Plane" is: 

The force required to raise a given weight over the 
plane is to the weight to be raised as the height of the plane 
is to the length of the sloping surface. 

To find the force required, multiply the weight on the 
plane by the height of the plane and divide by the length 
of the face of the plane. 

To find the weight which a given force will raise, 
multiply the force by the length of the face of the plane 
and divide by the height of the plane. 

These rules do not include the effect of friction on the 
plane. 

As shown in Fig. 77, a ^Svedge" consists of two inclined 
planes with their bases together. The force F used to 
drive the wedge acts parallel to the center line of the 
wedge. 

The principle of the wedge is the same as that for the 
inclined plane. In this case the wedge in being driven in 
is the same as if the weight being raised moved along a 
stationary inclined plane. Leaving out the effect of 



THE INCLINED PLANE AND WEDGE 



55 



friction on the sides of the wedge, the work done in driving 
the wedge equals the force exerted, F times I or FL 
The work done in raising the weight equals the weight 
lifted "W times the distance lifted "h'' or Wh. Then 
leaving out friction 

Fl = Wh or ^^ = -^or F = W X J 

Therefore the force required is only ^ times the weight 

lifted. 

Example. — In Fig. 78, a wedge 12 in. long has a taper 



Wcfgh-f 

/////7////y 




Fig. 77 




Fig. 78. 



of 13-^ in. per ft. and is used to raise a machine base 
weighing 1 ton. The force F required is found as 
follows : 

W Xh 2000 X 1.5 



F = 



I 



12 



250 lb. 



The ^^ mechanical advantage'^ in using the wedge is there- 
fore orrw or 8. That is, the force required is only 3^^ of 
the weight lifted. 



PROBLEMS 

1. A weight of 23^ tons is to be raised by a wedge having a 
taper of 1 in. per ft. What force is required to drive the wedge 
and what is the mechanical advantage leaving out the effect of 
friction ? 



56 



MECHANICS AND ALLIED SUBJECTS 



2. To wedge up a weight with a wedge tapering 2 in. per ft. 
requires a force of 500 lb. Neglecting friction, find the weight 
raised. 

24. The Screw Jack. — One of the most common ap- 
plications of the principle of the inclined plane is in the 
ordinary screw thread, which is nothing but an inclined 
plane wound around a cylinder. 

If we cut out a wedge-shaped piece from thick paper and 
wind it around a cylinder we will see that the sloping 
side forms a thread. The 
form of this thread as in Fig. 
79 is called a ^^ helix. ^' All 
nut, bolt, jack, and screw 
threads are circular or spiral 
wedges. 

Fig. 80 represents a screw 





Fig. 79. 



Fig. 80. 



jack which is used to overcome a heavy pressure or raise 
a heavy weight at TF by a much smaller force F applied 
at the handle. R represents the length of the handle 
and P the pitch of the screw, or the distance the screw 
advances in one complete turn. 

Neglecting friction the following rule is used : The force 
F multiplied by the distance through which it moves in 
one complete turn is equal to the weight lifted times the 
distance through which it is lifted in the same time. In 
one complete turn the end of the handle describes a 



THE INCLINED PLANE AND WEDGE 57 

circle of circumference 2TrR, This is the distance through 
which the force F is exerted. 
Therefore from the rule above 

F^X 2tR = W XP 

W X P 
T = 3.14 SLiidF = ' p 

Suppose in Fig. 80, R = 18 in., P = }y^ in. and the 
weight to be lifted is 50 tons or 100,000 lb., the force in 
lb. required at F is then found as follows: 

W XP 



F = 



2itR 



„ 100,000 X ^ 

^ = 2;^1^— ^^ 

^ _ 100,000 ^ 1 



8 ^ 27rl8 

Cancelling we get F = — t^ = ^^ ^ =110 lb. necessary. 

Try Zo.o 

This means then that neglecting friction 110 lb. at F 
will raise 100,000 lb. at W, but the weight lifted moves 
much slower than the force applied at F, 

PROBLEMS 

3. What weight can be held on a plane which rises 4 ft. in 
every 5 ft. by a force of 100 lb.? 

4. The height of an inclined plane is 10 ft. and its length is 
20 ft. What weight will a force of 25 lb. at F in Fig. 76 hold up 
on the plane? 

5. What counterbalance would be necessary to hold an empty 
coal car weighing 500 lb. on an incline 20 ft. long and 8 ft. high? 

6. A screw jack has a single thread 8 threads per in. and the 
handle is 2 ft. long. What force is necessary at F to raise a weight 
of 2500 lb., neglecting friction? 



58 MECHANICS AND ALLIED SUBJECTS 

7. What effect has increasing the pitch P, on the force neces- 
sary to raise a given weight if the length R does not change? 

8. It is desired to jack up a machine with a square base and 
weighing 5 tons by four jacks, one at each corner. If the jacks 
have 6 threads per in. and l)^2-ft. handles, and the weight is evenly 
distributed, what force is necessary at F? 

9. What effect has increasing the length of handle on the force 
F necessary to raise a given weight if the same pitch is used? 

10. A force of 50 lb. at F raises a weight of 3768 lb. at W with 
R = l}i ft. What is the pitch of the screw? 



CHAPTER VIII 

GEARS, LATHE GEARING 

25. Definitions. — A gear, or toothed wheel, when in 

operation, may be considered as a lever of the first class 

with the addition that it can be rotated continuously 

instead of rocking back and forth through a short 

distance. What we should learn here is the relation 

between the number of teeth, the diameter, and the 

speed of gears. Fig. 81 shows the ends of two shafts 

A and B connected by 2 gears 

of 24 and 48 teeth respectively. 

Notice that the larger gear 

will make only one-half turn 

while the smaller makes a 

complete turn. That is, the 2-^ TeeM 

ratio of speeds of the larger 

to the smaller is as 1 is to 2, ^ ^Tee^/? 

, . Fig. 81. 

or expressed as a proportion. 

Speed of B : Speed of A = 1:2 and the ratio between the 

speeds and the number of teeth written as a proportion is 

Speed of B : Speed of A - Teeth of A : Teeth of B. 
If in Fig. 81 gear A had 24 teeth and B 96 teeth, the 
smaller gear would make four turns while the larger 
makes 1, or. 

Speed of A : Speed of B = 96: 24, that is, the gear with 
the smaller number of teeth nmst turn the faster. 

Of two gears running together that one is called the 

59 




60 MECHANICS AND ALLIED SUBJECTS 

driver which is nearer the source of power and the second 
gear which receives power from the driver is called the 
follower. 

26. Gear Trains (Simple Gearing). — In the case of 
gear trains there may be several drivers and several fol- 
lowers. When the teeth on a gear turn in the same direc- 
tion as the hands of a clock the motion is called right 
handed or ^'clockwise/' and when in the opposite direc- 
tion, left handedj or ^^counter-clockwise/' In Fig. 82 
suppose the three gears A, 5, and C have 96, 24, and 96 
teeth respectively. 



S6T 967 




Fig. 82. 

When gear A turns once right handed, gear B turns 
4 times left handed and gear C turns once right handed. 
Hence gear B does not change the speed of C from what 
it would have been if geared directly to A, hut it changes 
its direction froin left handed to right handed. 

The ratio of speeds of the first and last gears in a train 
of simple gears is not changed by putting any number of 
gears between them. 

27. Gear Trains (Compound Gearing). — Fig. 83 shows 
^^ compound gears'' in which there are two gears on the 
middle shaft. Gears B and D rotate at the same speed 
since they are keyed to the same shaft. If the gears 
have a number of teeth as shown by the numbers in the 



GEARS, LATHE GEARING 61 

figure and gear A makes 100 r.p.m. right handed, gear B 
turns 400 r.p.m. left handed, also gear D turns 400 r.p.m. 
left handed and gear C turns 1200 r.p.m. right handed, 

28. Lathe Gears for Screw Cutting (Simple Gearing). 
— Modern lathes are equipped with a plate which gives 
the gears required for cutting different screw threads. To 
find the gears required in any case however it is merely 
necessary to understand the principle of a ^ ^simple'' or a 



^er 




24T 



100 R.p.m, 
Fig. 83. 

^'compound'' gear train depending on whether the lathe 
is simple or compound geared as explained in the 
following: 

Fig. 84 shows a ^^ simple geared'^ lathe with the names 
of important parts indicated. The lathe carriage carry- 
ing the cutting tool is moved by the lead screw which has 
usually 2, 4, 6, or 8 threads per in. If the lead screw 
has 4 threads per in., each complete revolution of this 
screw moves the carriage and hence the cutting tool Y^ 
in. along the work being threaded, and 4 turns moves the 
tool 1 in. along the work. If now we wish to cut 12 
threads per in., the spindle must revolve 12 times while 
the lead screw revolves 4 times. That is, 

turns of spi ndle _ 12 3 

turns of lead screw 4 1 



62 



MECHANICS AND ALLIED SUBJECTS 



The gears on the lead screw and on the spmdle must 
therefore have a ratio of teeth of 3 to 1, or 
teeth on lead screw gear _ 3 
teeth on spindle gear 1 

since the lead screw turns the slower. 



One of fwo reverse 
gears for changing 
direcfion orf rotctfion 
of gear"a" 

■P 



Driving Beli 










Ic//er(usec/ mere/t/-h 
connect gears "s^'andTf 



J Work be/n\ 
ihreadec 



t 



Lead Screw 



Fig. 84. 



If the gear a on the stud has the same number of teeth 
as gear p on the spindle, we have the rule 
teeth of lead screw gear _ 3 
teeth of stud gear 1 



GEARS, LATHE GEARING 63 

The idler gear (c) is used to connect gears S and I, and 
its number of teeth does not influence the speed ratio of 
the spindle and lead screw. The same is true of the 
gear connecting the spindle with the stud. 

The following is the rule: The number of threads to 
be cut per in. equals the product of the number of 
threads per in. in the lead screw with the number of 
teeth in each driven gear and divided by the product of 
the number of teeth in each driving gear. In the figure 
shown, if the lead screw has 4 threads per in., gear p on 
the spindle 24 teeth, gear a on the stud 48 teeth, gear >S 
36 teeth, and gear I 72 teeth, the threads cut per in. 

.4X48X72 ^^ 
equal -^4^3^ or 16. 

Using the letters on the gears to indicate their number of 
teeth we have the rule, if n = the number of threads cut 

per in. and 
d = the number of threads per 
in. of lead screw 

d X a X I 

n = — 

p X s 

For another example: 

ii d = threads per in. of lead screw = 2 

a = teeth in gear on stud = 48 

p = teeth in gear on spindle = 24 

s = teeth in outside gear on stud = 24 

I = teeth in lead screw gear = 72 

^, 2 X 48 X 72 ;^ ^^ ^ 

ihen n = or ^94 — ^ ^ threads per m. 



Suppose on the other hand we wished to find gears 
s'' and ''/'' to cut 8 threads per in. with a lead screw 



64 MECHANICS AND ALLIED SUBJECTS 

having 6 threads per in. we have from the rule, 

dXaXl ^l nXv, c u-u 

n = — ' and - = i-— ; — ' irom which 

p X s s a X a 

I ^ 8Xp _ 8 2> 
s 6 X a 6 a 

to find the ratio - we must choose gears p and a. If we 

o 

take p as 36 teeth and a 24 teeth, we have 

i = § 36 _ 2 
s ~ 6 ^ 24 ~ 1 

that is gears I and s must have a ratio of 2 : 1, hence we 
can put a 48-tooth gear on the lead screw and a 24-tooth 
gear as the outside gear on the stud and 8 threads per in. 
will be cut. 

29. Lathe Gears for Screw Cutting (Compound Gear- 
ing). — In the compound geared lathe, as shown in Fig. 
85, there are two change gears of different sizes on the 
spindle between the gear ^^s^^ on the stud and the gear 
^^r^ on the lead screw. The rule to be used in this case 
is as follows: 

The threads to be cut per in, equal the product of 
the threads per in. of the lead screw and the teeth in 
each of the driven gears, divided by the product of the 
number of teeth in each of the driving gears. Repre- 
senting the number of teeth of the gears by the letters on 
the gears we have, if ^'n^^ is the threads cut per in. and 
^*r' the number of threads per in. of the lead screw. 

dX aXlX e 
n = 



pX sXf 



GEARS, LATHE GEARING 



65 



For example, suppose the lead screw has 5 threads per 
in. and the gears have a number of teeth as follows: 
a = 28, p = 28, I = 72, / = 36, s = 24, e = 24, the 
number of threads cut per in. according to the rule 
becomes 



Driving Be/f 




^^^'Lead Screw 



Fig. 85. 



n 



dX aXlX e 5X28X72X24 



pXsXf 



28 X 24 X 36 



or 10. 



Working the other way round, if we wish to find the gears 
to use to cut a given thread we proceed as follows: 

5 



66 MECHANICS AND ALLIED SUBJECTS 

If d = 5 and the threads cut per in. are to be 14, we 

have 

5XaXlX e „ . . 

14 = — — ^^ J. -> trom this 

p X s Xf 

14 _ aXlX e 
5 ~ pXsXf 

The ratio of the gears must therefore be as 14 is to 5. 
Suppose by trial we take e = 28 and s = 30 teeth, we 
then have 

14 _ g X / X 28 14 X 30 _ a X I 
5 ~pX30X/^'* 5X28 ~ pXf 

.u . • ^ X I 3 ^, 

that IS — rr-j. = T> the remaining ratio. 

To provide this ratio we may take / = 24, Z = 72, and 
a and p even gears as, for example, 36 teeth each. 
We then have 

a X I _ 36 X 72 3 

p X / " 36 X 24 ^'' 1 

PROBLEMS 

1. If 8 threads are to be cut per in. and the lead screw has 6 
threads per in., what must be the ratio of revolutions for the spindle 
and lead screw? 

2. In a simple geared lathe calculate the size of change gears to 
use to cut 14 threads per in. if the lead screw has 5 threads per in. 

3. In a lathe the ratio of revolutions of lead screw and spindle 
is }i. If the pitch of the thread on the lead screw is J^ in., how 
many threads are cut for 16 revolutions of the lead screw? 

4. In a compound geared lathe calculate the size of change gears 
to use to cut 18 threads per in. if the lead screw has 4 threads per in. 

In working out the following problems think first of 
the relative sizes of the gears, which is the driver and 
which the follower, and remember that of two gears the 
larger must turn slower than the smaller. 



GEARS, LATHE GEARING 67 

The diameters of gears are proportional directly to 
their number of teeth and inversely proportional to their 
speeds. 

That is, if two gears A and B have pitch circles of 
diameters 12 and 24 in., respectively, and gear A has 
48 teeth, gear B must have 96 teeth and if the speed of 
A is 100 revolutions per minute (r.p.m.) the speed of B 
is 50 r.p.m. 

PROBLEMS 

5. If two gears A and B have 24 and 72 teeth respectively, 
and gear A turns right handed, in which direction will gear B turn? 
While gear A turns three times, how many times will B turn? 

6. If two shafts are to be connected by gears of 48 and 36 teeth, 
and the faster makes 900 r.p.m. what speed does the other make? 

7. Two gears have one 120 and the other 90 teeth. What is 
the ratio of the diameters of their pitch circles and of their speeds? 

8. Two gears have pitch circles 76 in. and 19 in. in diam. 
What is the ratio of their speeds? 

9. It is desired to run a shaft at 200 r.p.m. from another running 
at 500 r.p.m., what gears could be used? Specify by number of 
teeth? 

10. If 7 gears work in a train in what direction will the last one 
turn if the first turns left handed? How could you change the 
direction of rotation of the last gear? 

11. A train of gears is made up of 4 gears with the number of 
teeth as follows: 72, 48, 32, and 24. If the first gear makes 20 
r.p.m. right handed, in what direction and at what speed will the 
last gear move? 

12. A train of gears is made up of 5 gears with the number of 
teeth as follows: 48, 36, 24, 60, and 48. If the first gear makes 
30 r.p.m. left handed, in what direction and at what speed will 
the last gear move? 

13. A main shaft runs at 150 r.p.m. and carries a 96-tooth gear 
which drives a 24-tooth gear on a second shaft. A 72-tooth gear 
on the second shaft drives a 24-tooth gear on a third shaft. 
Find the speed of each shaft and make a sketch showing the ar- 
rangement of gears and directions of rotation produced. 



68 



MECHANICS AND ALLIED SUBJECTS 



14. A simple gear train has four gears with the number of teeth 
as follows: 96, 24, 48, and 60. If the last gear is on a shaft making 
500 r.p.m. and turns right handed, find the speed and direction 
of rotation of the first gear. 

15. According to Fig. 86, in what direction will gear 2 turn? 
Find the speed ratio of the gears. 

''Rafioliol 





Fig. 86. 



Fig. 87. 



42Teefh 



lOSTeeih 





Fig. 88. 



Fig. 89. 



16. What is the direction of rotation of gear 2 according to 
Fig. 87? Find the number of teeth in gear 2. 

17. Fig. '^'^ shows a train of simple gears. In what direction will 
gear 2 turn? Find the speed ratio of gears 1 and 2. 

18. Fig. 89 shows a train of compound gears. In what direction 
will gear 4 turn. 



CHAPTER IX 

BELTS AND PULLEYS. EFFICIENCY OF 
MACHINES 

30. Belts and Pulleys. — Belts and pulleys are an im- 
portant part of shop equipment and it is necessary to 
understand how to calculate speeds and diameters of 
pulleys and also lengths of belting. 

Pulleys are nothing but gears without teeth and in- 
stead of running together directly they are made to drive 
one another by cords, ropes, cables, or belting of some 
kind. 

As with gears, the speeds 
of pulleys are inversely pro- 
portional to their diameters. 
That is, if, as in Fig. 90, we 
have two pulleys A and B -pio. 90. 

which are 24 in. and 12 in. in 

diam. respectively, the larger pulley travels the slower and 
the speed of A is to the speed of B as 12 is to 24. 

This difference between gears running together di- 
rectly and pulleys driven through belting should be 
fully understood. A belt ^^ creeps ^^ and ^^ slips ^^ and does 
not run the driver quite as fast as would be figured out 
from the diameters of the pulleys, while gears cannot 
slip and the practical speed and calculated speeds are 
the same. 

When two pulleys are driven through a belt if there is 
no slipping of the belt, the rim speed of both pulleys is 

69 





70 MECHANICS AND ALLIED SUBJECTS 

the same and in Fig. 90 when pulley A makes 1 turn the 
belt passes over each pulley a distance equal to the 
circumference of pulley A or 27rl2 or 247r in. and if the 
rim of pulley B has passed through 247r in. for 1 turn of 
pulley A and its diameter is 12 in. it has made a number 

247r 
of turns in this time equal to z-^ or 2. The following 

simple rule is convenient for calculating speeds of pulleys. 
The diameter of the driving pulley multiplied by its 

speed is equal to the diameter 

of the driven pulley multiplied 

by its speed. In calculating 

the length of belt to connect 

two pulleys if the pulleys are 

the same size and the belting 

Fig. 91. _is not crossed the length of 

belt required is equal to twice 

the distance between the centers of the pulleys plus the 

circumference of one of the pulleys. In Fig. 91 the length 

of belt needed equals 

18 
2 X 6 + 7r Xj2 

or 12 + 4.7 or 16.7 ft. 

If the diameters of the pulleys are not the same, but do 
not differ greatly and the distance between their centers 
is large compared with the pulley diameters, the length 
of belt is found very nearly by adding to twice the dis- 
tance between the pulley centers half the circumference 
of each of the pulleys. 

PROBLEMS 

1. It is desired to run a countershaft at 200 r.p.m. from a 
line shaft running at 800 r.p.m. If the pulley on the line shaft is 
18 in. in diam., what size pulley is required on the countershaft? 



1 



BELTS AND PULLEYS 



71 



2. What length of belt is required to connect two 24-in. pulleys 
whose centers are 6 ft. 4 in. apart? 

3. A belt runs from a 36-in. pulley on a line shaft to a second 
shaft having a 14-in. pulley. If the line shaft makes 650 r.p.m. 
what is the speed of the second shaft? 

4. Find the length of belt required to connect a 24-in. and a 
36-in. pulley whose centers are 8 ft. apart. 

5. A main shaft running at 150 r.p.m. with an 18-in. pulley 
drives a jack shaft through a 12-in. pulley. A second pulley of 
24 in. diam. on the jack shaft drives a dynamo through a 12- 



^S:^^^<<<gg^ 



Laihe 
Counfershaff] 




Elecinc 
Motor 
5Hp. 
Speed SOO I Q\ Laihe 



Grinder 



Fig. 92. 

in. pulley. Find the speed of the dynamo and make a sketch 
showing the arrangement of pulleys, and direction of rotation 
of shafts. 

6. Calculate from the following rule the length of belting re- 
quired to connect two 24-in. pulleys 7 ft. apart if the belt is crossed? 

I = 2 Vz)2 + ^2 _|. ^O 

When D = diam. of pulleys 

S = distance between centers 

7. A main shaft running at 200 r.p.m. with a 12-in. pulley 
drives a second shaft through a 24-in. pulley. The slip of the belt 
is 2 per cent, of the driving speed. On the second shaft is a second 
pulley 36 in. in diam. which is connected through a belt to an 
18-in. pulley on an electric generator. The belt on the generator 



72 



MECHANICS AND ALLIED SUBJECTS 



slips 3 per cent, of the driving speed of the second shaft. What is 
the speed of the generator? 

8. Fig. 92 shows the elevation of a layout for a small machine 
shop, with the equipment as indicated. Working from this sketch, 
find 



SO Diarn 



Mo-tor R.p.m--I800 



Counter 
Shaft 




Fig. 93. 



1. The diameter of pulley on motor shaft. 

2. The speed in r.p.m. of lathe countershaft. 

3. The speed of lathe. Pulley on lathe countershaft is 1 2 in . 
in diameter. 

4. The speed of grinder countershaft. Pulley on grinder 
countershaft is 8 in. in diameter. 

5. The speed of grinder. 

6. The weight in pounds of line shaft. Length 10 ft., 
diam. 2>^ in. Material, steel weighing .28 lb. per cu. in. 



BELTS AND PULLEYS 73 

9. With the belts as arranged in Fig. 93, show the direction of 
turning of the lathe spindle, with the direction of rotation shown 
for motor. How could the direction of turning of the lathe be re- 
versed? With data given on sketch what is the speed in r.p.m. of 
the lathe spindle when belted as shown? Why is it considered 
better practice to have the tight side of the belt on the bottom of 
the pulleys? 

31. Efficiency of Machines. — In working out the prob- 
lems on levers, pulleys, inclined plane and so forth we 
have not taken account of friction or other sources of 
loss. In other words we have supposed them to be 
perfect. As actually used their operation is not as 
satisfactory as our answers show, and as an indicator 
of the performance of a machine we often find its 
*' efficiency.'^ 

the output of a machine 



Efficiency means 



or better efficiency = 



input to the machine 
output 



input 



The efficiency is a number which shows how much of the 
energy the machine takes, it puts out again as useful 
work. 

Suppose a screw jack has an efficiency of 20 per cent, 
or .20. This means that it delivers in useful work 20 per 
cent, of all the energy or work put into it. 

If a machine had an efficiency of 100 per cent, or 1 

it would mean that efficiency = 1 = . ^ therefore 

mput 

input X 1 = output, or output = input. But no ma- 
chine has an efficiency of 100 per cent, and therefore the 
output can never equal the input since there are always 
some losses in the machine itself. 

Suppose a 50-h.p. motor has an efficiency of 85 per 



74 MECHANICS AND ALLIED SUBJECTS 

cent, when delivering its rated (50) horsepower. To 
find its input we have 

Efficiency = ~ — , therefore 

output . 50 c,o o u 

input = -^. or input = -^ or 58.8 h.p. 

If a steam engine deUvers 50 h.p. and its input is 70 
h.p., its efficiency is found as follows: 

Eff. = ^^ = 1-^ = .714 
input 70 

or .714 X 100 = 71.4 per cent. 

It is therefore to be remembered that in working out a 
problem on machines we always have to take account of 
losses that are due to friction in the machines them- 
selves in order to get the correct results. 

PROBLEMS 

10. An electric generator delivers 48 h.p. and requires 60 h.p. 
to drive it. What is its efficienc}^? 

11. A screw jack lifts a 20-ton weight 2 ft. thereby doing 
20 X 2000 X 2 or 80,000 ft.-lb. of work, and in doing this work 
requires 334,000 ft.-lb. of work. Find its efficiency. 

12. An engine is rated at 100 h.p. and has an efficiency at this 
load of 75 per cent. What is its input? 

13. An electric motor takes 20 h.p. to do a certain amount of 
work. If its efficiency is 88 per cent., find its horsepower output. 



CHAPTER X 
MOTION 

32. Definitions.— Motion is a change of position and 
may be of a number of kinds. 

In order to set objects in motion forces or pressures are 
necessary. If a steady force acts on an object moving 
it, the object increases in motion until the resistance 
with which the object meets is equal to the force causing 
the motion and the object then travels over equal dis- 
tances in equal times or as we say, has a constant speed. 
The word constant means unchanging, and speed, or 
what is the same thing, velocity, means the rate of motion 
of an object. If a train going at a steady rate covers 
fifteen miles in an hour its speed or velocity is 15 miles 
per hour (abbreviated m.p.h.). When an object is mov- 
ing at a constant speed, if the driving force is removed, 
the speed decreases until the object is brought to rest by 
resisting forces such as friction, etc. When an object 
increases in speed it is said to ^^ accelerate,'^ and when it 
decreases in speed it is said to ^^ retard. '' 

Any object at rest will remain at rest for all time and 
an object in motion will always move at a constant speed 
in a straight line unless it is acted upon by a force or 
pressure which deflects it or changes its rate of motion, 
and how much an object will change in speed or direction 
or both depends on how large the force is acting on it. 

When an object is free to fall the force of attraction of 

75 



76 MECHANICS AND ALLIED SUBJECTS 

the earth for it or the force of gravity causes the object to 
gam in speed 32.2 ft, per sec. each second of its fall, 

33. Composition of Motions. — Motions, like forces, 
may be conveniently represented by straight lines. The 
length of the line representing to scale the rate of the 
motion as 10 ft. per sec. or 40 m.p.h. and the direc- 
tion of the line, the direction of the motion. In Fig. 
94 an object at is represented as having two successive 
motions of 10 and 15 miles per hour, these motions be- 
ing 60° apart. The effect of the two motions is the same 
^ as one motion of 22 m.p.h. in the di- 

J I'y^^ rection OC. This occurs, for exam- 
.^/ ^^^y^/ pl^? if from a train traveling in the 

Xv4/^ / direction OB at 10 m.p.h., an object 

7/^0^ / is thrown off in the direction OD at 

^ lOM.P.H. a speed of 15 m.p.h., the object itself 

Fig. 94. ^^ affected by both motions at the 

same time and travels in the direction 
OC at the rate of 22 m.p.h. If two trains are travel- 
ing in the same direction, one at 40 m.p.h. and the 
other at 35 m.p.h. the relative speed of the trains is only 
5 m.p.h. 

If an engine hauls a freight train at the rate of 20 m.p.h. 
and a brakeman runs along the train in the direction of 
motion at a speed of 5 m.p.h. he is traveling at the rate 
of 20 + 5 or 25 m.p.h. with respect to the earth or track. 

The following rules are convenient in determining 

speeds. 

^^ , ., , distance 

Velocity or speed = ——. • 

^ ^ time 

and distance = speed X time 

speed 



Acceleration or rate of change of speed = 



time 



MOTION 77 

If a train starts from rest and increases uniformly in 
speed to 24 m.p.h. in 3 min. it has changed in speed or 

24 
''accelerated^^ ^ or 8 m.p.h. each minute. If this con- 
stant rate of increase of speed were kept up for 5 min. 
by a locomotive starting from rest, its speed at the end of 
the 5 min. would be 8 X.5 or 40 m.p.h. 

It has already been stated that the earth attracting a 
freely falling object increases its speed about 32.2 ft. 
per sec. for each second of its fall. 

PROBLEMS 

1. A locomotive went from Altoona, Pa. to Harrisburg, Pa., a 
distance of 131 miles in 112 min. What was its average speed in 
miles per hr., and in ft. per sec? 

2. The same locomotive mentioned in Prob. 1 went from Al- 
toona to Philadelphia, a distance of 236 miles in 3 hr. and 19 min. 
Making allowance for a 3-min. stop at Harrisburg, find the average 
speed made in miles per hr. and in ft. per sec. 

3. A passenger locomotive leaving a station attains a speed of 
40 m.p.h. in 4 min. 30 sec. What is the acceleration or rate of 
change of speed? 

4. An engineer has to make, in 3 hr., a 120-mile run with four 
3-min. stops. W^hat average rate of speed does he have to main- 
tain in order to complete the run on time? 

5. A passenger train making an average speed of 38 m.p.h. has 
to make a run of 85 miles. How much sooner will it complete 
the run than a freight which starts at the same time and makes an 
average of 25 m.p.h.? 

6. The '' Pennsylvania Limited" traveling at 60 m.p.h. starts 
from a station an hour after an accommodation train. The 
accommodation is delayed 3^^ hr. and is overtaken by the ^'Limited 
in 2 hr. What is the rate of speed of the accommodation train 
and how far had it traveled when overtaken by the '* Limited?" 

7. If the 82-in. drivers on a passenger locomotive make 200 
revolutions per minute (r.p.m.) what is its speed in miles per hr.? 
What would be its speed allowing for a 2 per cent, slip? 



78 MECHANICS AND ALLIED SUBJECTS 

8. A passenger locomotive with 80-in. drivers making 180 
r.p.m. and a freight engine with 60-in. drivers making 150 r.p.m. 
are traveUng in the same direction. What is the relative speed 
of the locomotives? If the locomotives were traveling in opposite 
directions what would be their relative speeds? 

9. An engineer makes an average of 42 m.p.h. and runs for 4 hr. 
and 20 min. What distance does he cover? 

10. If an object falls freely from rest for 3 sec, what is its speed 
in ft. per sec. and in miles per hr. at the end of that time? 

11. The drivers on a locomotive are 18.85 ft. in circumference 
and make 250 turns a min. What is the speed of the locomotive 
in miles per hr. ? 

12. Two trains start from a station at the same time and travel 
over level country, the first North at the average rate of 40 m.p.h. 
and the second East at the average rate of 30 m.p.h. How far 
apart are the trains at the end of 5 hr. ? 

13. A train makes a total distance of 236 miles, covering one- 
third of this distance at the rate of 45 m.p.h. and the remaining 
distance at the rate of 40 m.p.h. In what time does it make the 
trip if its total stops take 10 min.? 

14. A touring car starts from a garage and reaches a speed of 
48 m.p.h. in 4 min. and 20 sec. What is its acceleration in miles 
per hr. per min., and in miles per hr. per sec? 

15. Find the speed in ft. per sec. and in miles per hr. of a freely 
falling object at the end of the fourth second of its fall. 

16. In a trolley car the gear on the driving axle and the gear 
on the motor axle have a ratio of teeth of 4:1. The driving 
wheel has a diameter of 32 in. Find the speed of the trolley car 
in miles per hr., when the motor is making 1000 r.p.m. 

17. Two trains start from a station at the same time, one going 
East at 40 m.p.h. for 5 hr. and the other North at 20 m.p.h. for 
2 hr. and then East at 10 m.p.h. for 3 hr. Find the distance apart 
of the trains at the end of the 5 hr. 



CHAPTER XI 
CUTTING SPEEDS. SPEEDS OF LATHES 

34. Definitions and Problems. — The calculation of 
the circumference and diameter of circles becomes neces- 
sary in calculating the cutting speed of work being 
turned off in a lathe or in calculating the rim speed of 
fly-wheels, grindstones, etc. 

Machines, such as lathes, milling machines, boring 
mills, planers, etc., are provided with means for changing 
-the speed according to the work turned out. Every 
mechanic should know how to calculate the proper size 
of pulleys or gears for any particular job. When a piece 
of metal of circular section is being turned down in a 
lathe the cutting tool for each revolution of the work 
travels a distance (with respect to the metal being 
turned down) equal to the circumference of the work. 

For example, a rod 2 in. in diam. passes a distance 
of 3.142 X 2 or 6.284 in. under the cutting tool for each 
revolution of the rod. If the rod rotated 60 times per 
min. the cutting speed would be 60 times 6.28 or 376.8 

376.8 ^^ ^ „^ 
m. per mm. or ^ or 31.4 ft. per mm. 

If S is the surface speed in ft. per min., 
N the revolutions per min. of the work, 
C the circumference of the work in in., 

the rule is S = — ^o — 

79 



80 MECHANICS AND ALLIED SUBJECTS 

For example, a 2-in. diam. brass rod when turned at the 
rate of 160 r.p.m. has a surface speed of 

2 X 3.142 X 160 ___„^ 

The rim speed of a fly-wheel 72 in. in diam. when 
running at 200 r.p.m. may be found by the same rule as 
follows : 

^ 72 X 3.142 X 200 __ ,^ 

o = y^ or 3770 it. per mm. 

Working the other way round we can find the revolu- 
tions per min. (A^) at which a wheel must rotate to have 
a given surface or rim speed. 

If a fly-wheel 48 in. in diam. is to have a rim speed of a 
mile a min. or 5280 ft. per min. its revolutions per min. are 
found from the rule 

.. 12X>S ,, , . 

-^ , ,. . 12 X speed in ft. per min. 

Revolutions per mm. = —. /- -^ — , , . — -. — 

circumference oi wheel in in. 

rru f AT 12 X 52 80 ^^^ 

Therefore N = 43 ^ 3 ^42 = ^20 r.p.m. 

There are average values given for cutting speeds of 
different metals as follows: 

Cutting Speeds in Ft, per Min. 

Brass 80 

Machinery steel 30 

Tool Steel 20 

Cast Iron 40 

With high-speed steel cutting tools about double the 
above speeds can be used. No exact figures can be given 
for the best cutting speeds of the different metals. The 



CUTTING SPEEDS 81 

proper speed to use depends on the size of the work, the 
power and rigidity of the lathe, the kind of tool used and 
the nature of the work being turned out. 

For the surface speed of grindstones the following fig- 
ures may be taken 

For carpenters' tools, 600 ft. per min. 
For machinists' tools, 1000 ft. per min. 

When edge tools are ground care must be taken to 
avoid too high a speed to prevent heating the edges of the 
tools and getting rough surfaces. For rapid grinding the 
following speeds are used: 

With Huron stones, 3500 ft. per min. 
With Ohio stones, 2500 ft. per min. 

Some polishing wheels are run as high as 8000 ft. per min. 
Emery wheels are run usually at a surface speed of about 
a mile a min. or 5280 ft. per min. 

To find how long it will take the cutting tool on a lathe 
to travel a given distance along the work with a given feed 
we proceed as follows: 

Suppose a lathe tool has a feed of }i in. per rev., and we 
wish to find how long it will take to turn a length of 18 
in. on an 8-in. diam. piece with a cutting speed of 60 ft. 
per min. The circumference of the piece is 8 X 3.142 or 

25.136 in. This is equal to ^^ or 2.094 ft. Withacut- 

ting speed of 60 ft. per min. this requires ^ ^^ . or 28.6 r.p.m. 

For each revolution with a feed of 3^^ in. the tool travels 
along the work 3^^ of an in. or 28.6 X H = 3.57 in. per 

min. To turn a length of 18 in. would therefore require 

18 
^-ry or 5.04 min. or a little over 5 min. 

6 



82 



MECHANICS AND ALLIED SUBJECTS 




^ Jr ® ?. N li! 






oi 




P 




c3 




e3 




OJ 








^ 




fl 




HH 




tT 




fl 


'9 
«7 




i. 




& 


^ 


A 


^-s 


B 


:3 
o 


3 


rr> 


a 






CO 


a 


^ 




;m 


k 


O 


CI 

a 


^ 


^ 


CD 


a 


^ 


A 


^J 




Cj 


B 


h:j 


"Z 


^ 


a 


j^ 




CJ 


Ul 


PQ 


a 


^ 


^ 


:3 


e 


o 


«9 


02 



CUTTING SPEEDS 



83 



NUMBER AND NAME OF LATHE PARTS ON DRAWING 



NO. 

1. Bed. 

3. Power Legs. 

4. Lead Screw Bracket F. 

5. Lead Screw Bracket R. 

10. Head Stock. 

11. Head Stock Cap, Large. 

12. Head Stock Cap, Small. 

13. Head Stock Clamp Plate. 

14. Spindle Cone. 

15. Bull Gear. 

16. Bull Gear Clamp. 

17. Cone Pinion. 

18. Quill Gear. 

19. Quill Sleeve. 

20. Quill Sleeve Pinion. 

21. Ecc. Shaft Bushing. 

22. Bronze Box, Large. 

23. Bronze Box, Small. 

24. Back Gear Lever. 

25. Spindle Take up Nut. 

26. Reverse Bracket. 

27. Reverse Twin Gears (2). 

28. Reverse Gear. 

29. Stud Gear. 

30. Spindle Reverse Gear. 

31. Change Gear Bracket. 

32. Change Gears. 

33. Change Gear Idler. 

34. Bushing for Change Gear Idler. 

35. Change Gear Collar on L. S. 

36. Compound Idler Gear 1 to 2 
Large. 

37. Compound Idler Gear 1 to 2 
Small. 

38. Bushing for Compound Idler 
Gear, 1 to 2. 

39. Thrust Collar on Lead Screw. 

40. Large Face Plate. 

41. Small Face Plate. 

42. Turning Gear. 

50. Tail Stock Top. 

51. Tail Stock Base. 

52. Tail Stock Nut. 

53. Tail Stock Hand Wheel. 

54. Tail Stock Binding Lever. 

55. Tail Stock Wrench. 



56. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
70. 
71. 
72. 
73. 
74. 
75. 
76. 
77. 
78. 

79. 
80. 
81. 
82. 
83. 
84. 
85. 
86. 
87. 
90. 
91. 
92. 
93. 
94. 
95. 
96. 



Tail Stock Clamp Plate. 

Saddle. 

Saddle Gib. 

Saddle Lock. 

Cross Feed Bushing. 

Cross Feed Gra. Collar. 

Cross Feed Nut. 

Plain Rest. 

Thread Cutting Stop. 

Apron. 

Apron Hand Wheel. 

Lead Screw Half Nut. 

Lead Screw Half Nut Gib (2). 

Nut Cam. 

Nut Cam Washer. 

Rack Pinion Gear. 

Auto. Apron Worm Wheel. 

Apron Clutch Sleeve Bush- 



Auto 
ing. 
Auto 
Auto 



Apron Worm Bracket. 

Apron Clutch Sleeve. 
Auto. Apron Clutch. 
Auto. Apron C. F. Star Knob. 
Auto. Apron C. F. Lever. 
Auto. Apron C. F. Lever Knob. 
Auto. Apron C. F. Gear. 
Auto. Apron Idler Gear. 
Auto. Apron Idler Gear Pinion. 
Compound Rest Top. 
Compound Rest Swivel. 
Compound Rest Bottom. 
Compound Rest End Cap. 
Compound Rest Bushing. 
Clmpound Rest Nut. 
Compound Rest Chip Guard. 

Countershaft 



100. C. S. Friction Pulleys (2). 

101. C. S. Friction Spiders (2). 

102. C. S. Friction Fingers (2). 

103. C. S. 

104. C. S. 

105. C. S. 

106. C. S. 

107. C. S. 



Friction Spiders 
Friction Fingers 
Cone. 

Collars (4). 
Yoke Lever. 
Boxes (2). 
Hangers (2). 



84 



MECHANICS AND ALLIED SUBJECTS 



NO. 




NO. 


108. 


C. S. Shipper Nut. 


230. 


109. 


C. S. Yoke Cone. 


231. 


200. 


Head Stock Spindle. 


232. 


201. 


Tail Stock Spindle. 


233. 


202. 


Back Gear Eccentric Shaft. 


234. 


203. 


Apron Worm. 


235. 


204. 


Apron Rack Pinion. 


238. 


205. 


Spindle Sleeve. 


239. 


207. 


Spindle Thrust Collar. 


240. 


208. 


Apron Worm Collar. 


241. 


209. 


Tool Post Block. 


242. 


210. 


Carriage Lock Collar Screw. 


250. 


211. 


Compound Rest Swivel Bolts. 


251. 


212. 


C. G. Bracket Collar Screw. 


252. 


213. 


Reverse Collar Screw. 


253. 


214. 


Bull Gear Clamp Collar Screw. 


254. 


215. 


Apron Clutch Sleeve Hex. Nut. 


257. 


216. 


Compound Rest Swivel Stud. 


258. 


217. 


Steady Rest Lock Bolt. 




218. 


Auto Cross Feed Lever Stud. 


260. 


219. 


Reverse Steel Washer y/' hole. 


261. 


220. 


Apron Clutch Sleeve Pinion. 


262. 


221. 


Compound Rest Bottom Gib. 


263. 


222. 


Plain Rest Gib. 


275. 


223. 


Auto Apron Clutch Screw. 


224. 


Cross Feed Screw. 


276. 


225. 


Apron Hand Wheel Pinion. 


277. 


226. 


Tail Stock Screw. 


278. 


227. 


Reverse Shaft or Stud. 


279. 


228. 


Apron Rack Pinion Stud. 




229. 


Reverse Shoulder Screws (2). 


300. 



Compound Rest Screw. 

Auto Cross Feed Stud. 

Apron Half Nut Stud (2). 

Tool Post Screw. 

Apron Idler Gear Stud. 

Cam Cap Screw. 

Apron Worm Washer. 

Comp. Rest Steel Wedge. 

Gap Bridge Pins (2). 

Reverse Stud Collar W hole. 

Change Gear Spindle Knob. 

Tool Post. 

Tool Post Ring. 

Tool Post Wedge. 

Tool Post Wrench. 

Compound Rest Wrench. 

Compound Rest Top Gib. 

Comp. Rest C. P. Headless Set 

Screws. 

Centers (2). 

C. S. Shaft. 

C. S. Shipper Rod. 

C S. Expansion Wedges. 

Rack. 

Cross Feed Ball Crank. 

Compound Rest Handle. 

Tail Stock Set Over Screws (2). 

Tail Stock Clamping Bolt, Nut 

and Washer. 

Lead Screw. 



PROBLEMS 

» 

1. A steel bar 3 in. in diam. is to be turned in a lathe. If the 
cutting speed is to be 30 ft. per min. at how many revolutions 
per min. should the bar turn? 

2. A cast-iron pulley 18 in. in diam. is to have its rim face 
turned down in a lathe. With a cutting speed of 40 ft. per min. 
what should b,e the revolutions per min. of the pulley? 



CUTTING SPEEDS 



85 



- .r'l,' ;'■■,'■, 4 


— ^-T'c-i'i-.'.'l 


iji i ■' • . :.• ■," J v'n 'y'y«,it^MU'i.^> i'''wj,''i<f >'''': • 








\f '^^^j 


Pjl^ 


\...^ 


F 



Combination Square 




Micrometer Caliper 



i'ir'ri''V'V'i'''i'''i'''r''i'''!''V''i''''i'''i'''i'''i''i'''i'''Fn'''iLi 



Steel Rule 




Protractor 




NO. 145 

TAKES 8 IN.-n} 12 IHSAWS 




Hack Sai 




Divider 




Caliper 



ArHouuas as> 



Steel Square 



Vt 




^ 




ilMliliWllliiiilillMillliiliMllli 
Depth Gage 




im 



N2257 




Surface Gage 



Fig. 96 — Cut showing machinist tools. 



86 MECHANICS AND ALLIED SUBJECTS 

3. An emery wheel 14 in. in diam. runs at 1380 revolutions per 
minute (r.p.m.). Find its surface speed in ft. per min. 

4. What is the speed of a belt running on a 3-ft. pulley which 
makes 300 r.p.m.? 

5. At how many revolutions per min. should we run a 2'in. 
diam. brass rod in a lathe for a cutting speed of 80 ft. per min.? 

6. Find the revolutions per min. for a l)^-in high-speed drill 
to give a cutting speed at the outer edge of 80 ft. per min. If the 
feed is .015 in. per rev., how long will it take to drill through 13^^ 
m. of metal? 

7. Find the revolutions per min. of an 18-in. polishing wheel 
whose surface speed is 8000 ft. per min. 

8. At how many revolutions per min. should a 30-in. grind- 
stone be run to obtain the proper surface speed for grinding 
machinists' tools? 

9. A piece of work 6 in. in diam. is being turned down at a 
cutting speed of 60 ft. per min. (f.p.m.). If the cutting tool feeds 
along the work }<§ in. for each revolution, how long will it take to 
turn a length of 8 in.? 

10. A steel crank shaft is to be finished to a diameter of 8 in. 
With a cutting speed of 40 f.p.m. what should be its speed in revo- 
lutions per min.? 



CHAPTER XII 

VOLUME AND PRESSURE OF GASES 

35. Definitions and Problems. — A very important law 
has been discovered in regard to the relation of the vol- 
ume, temperature and pressure of gases. Steam follows 
this law very closely. It has been found that if the tem- 
perature of a gas does not change, its volume is practically 
*' inversely proportional'' to its pressure, or its volume 
times its pressure is a constant quantity. This means 
that if we have a cylinder, as in Fig. 97, which contains 
10 cu. ft. of air with the piston in 
position A and the gauge reads O^zzzzzz^z^zz^ 
10 lb., making an absolute pres- VJ ^ • '—-"■ 




sure of 10 -j- atmospheric pressure ^^^^^^^}^}^^^r^u\ (^ 

(that is, 10 + 14.7 or 24.7 lb.), and p^^ g^^ 

we move the piston to the posi- 
tion B so that the volume is only 5 cu. ft. or one-half of 
its former value, the absolute pressure becomes 2 X 24.7 
lb. or 49.4 lb. or twice its former value and the gauge 
reads 49.4 — 14.7 or 34.7 lb. since the gauge is made to 
read pressures above that of the atmosphere. That is, 
making the gas occupy half the volume has doubled its 
pressure. If, on the other hand, we increase the volume 
of the gas, we lower its pressure. This is exactly what 
happens in a steam engine cylinder into which steam en- 
ters at perhaps 200-lb. pressure, then expands, does work 

87 



88 MECHANICS AND ALLIED SUBJECTS 

in moving the piston, becomes reduced to a compara- 
tively low pressure and exhausts into the air. 

The principle of expansion and compression of gases 
is made extensive use of in all steam and gas engines and 
in air-braking devices. The rule is: If the temperature 
of a gas does not change, the product of its volume and 
pressure remain practically the same throughout a con- 
siderable range of pressure. 

PROBLEMS 

1. If the air in a cylinder containing 5 cu. ft. at atmospheric 
pressure is compressed to a volume of 1 cu. ft., what is its pressure? 

2. A gas occupies 10 cu. ft. at an atmospheric pressure of 15 lb. 
per sq. in. and is afterward compressed to a volume of 2 cu. ft. 
What is its pressure? 

3. During the movement of a piston in a steam cylinder the 
pressure drops from 90 lb. to 50 lb. If this change took place 
according to the law explained, what change of volume took 
place? 

4. A 30-in. X 36-in. cylinder with a .2 cu. ft. clearance volume 
at each end takes in air at atmospheric pressure of 14.7 lb. What is 
the pressure of this air when the piston has completed one-half 
the return stroke and is compressing the free air drawn in on the 
forward stroke? 

5. Twenty cubic feet of free air is compressed to a pressure of 
60 lb. gauge. What volume does it then occupy? 

6. The air cylinder of an air compressor is 20 in. X 28 in. and 
its clearance volume is .2 of a cu. ft. What is the pressure of 
the discharged air if the outlet valves lift at ^i stroke? 

7. If the admission of steam to a 24-in. X 26-in. cylinder at a 
pressure of 200 lb. per sq. in., gauge is cutoff at 50 per cent, of the 
piston stroke (H stroke) what is the terminal pressure, that is, 
the pressure at the end of .the stroke?. (Give the answer in gauge 
pressure.) Clearance volume at each end of cylinder is .2 cu. ft. 

8. What would be the terminal pressure if the cut-off occurs 
at 25 per cent, of the stroke {}i stroke) with the conditions of 
Prob. 7? 



VOLUME AND PRESSURE OF GASES 89 

9. With a back pressure on the piston during the exhaust stroke 
of 3 lb. per sq. in. gauge, what would be the compression pressure 
(lb. per sq. in. gauge) at the end of the stroke when compression or 
exhaust closure occurs at .6 of the piston stroke, in a 24-in. X 
26-in. cylinder? Clearance volume at end of cylinder is .2 cu. ft. 

10. What would be the compression pressure (lb. per sq. in. 
gauge) if compression of the steam occurs at % stroke with other 
conditions the same as in Prob. 9? 



CHAPTER XIII 
WORK AND POWER. 

36. Definitions and Rules on Work. — Work means the 
overcoming of resistance of any kind. If you drag an 
object along the floor you do work in overcoming the 
friction between the object and the floor. In lifting an 
object you do work against gravity which tends to pull 
the object toward the earth. Steam in a locomotive 
cylinder does work when it expands and moves the piston 
against the resisting forces. ^^ Work '^ is measured in ^^foot- 
pounds/' that is, the product of the resistance overcome in 
lb. and the distance in ft, through which it is overcome. If 
you lift a 10-lb. weight 5 ft. you do 10 X 5 or 50 ft. -lb. 
of work against gravity. 

If a train weighs 350 tons and the resistance offered 
to its motion is 12 lb. per ton, the work done in moving 
the train 1 mile at uniform speed is 

350 X 12 X 5280 = 22,176,000 ft.-lb. 

If it takes an average pressure of 30 tons to punch a 
^^-in. steel plate, the work done equals the resistance in 
lb., or 30 X 2000 lb. times the thickness of the plate in 

ft., % in. = ys2 ft. 

Therefore work done = 30 X 2000 X M2 

= 1875 ft.-lb. 

37. Power, Horsepower. — Power means the rate at 
which work is done and is expressed in foot-pounds per 

91 



92 MECHANICS AND ALLIED SUBJECTS 

min. or in foot-pounds per sec. When 550 ft. -lb. of work 
are done in a sec. or 33,000 ft.-lb. in a min. 1 h.p. is rep- 
resented. A 10-horsepower (abbreviated h.p.), engine 
therefore is one which can do 33,000 X 10 or 330,000 
ftAb. of work in 1 min., or is one capable of overcoming 
a resistance of 330,000 lb. through a distance of 1 ft. in 
1 mm. 

The rule for horsepower is: 

T-r pounds X feet 

Horsepower = ^.^^ ^^ ;- 

550 X seconds 

T-r pounds X feet 

Horsepower = spOO X minutes 

In electrical terms 1 h.p. is equal to 746 watts. The 
horsepower required in any case depends as much on the 
time in which the work is to be done as on the amount of 
work to be done. 

PROBLEMS 

1. What is the horsepower of an engine that will raise 8000 lb. 
of water 200 ft. in 5 min.? 

2. What is the horsepower of an automobile engine which can 
do 82,500,000 ft.-lb. of work in 5 min.? 

3. What horsepower motors are required to raise a 222,000-lb. 
locomotive 4 ft. in 1 min. ? 

4. If a man in a day of 8 hr. raises 20,000 lb. of gravel through 
an average distance of 5 ft., what horsepower represents his work- 
ing power? 

5. A man weighing 150 lb. runs up stairs to a vertical height 
of 20 ft. in 2 min. What horsepower is he exerting to raise his 
own weight? 

6. What is the useful horsepower of an engine which draws a 
train of 200 tons on a level at the uniform rate of 30 miles per hr. 
against a resistance of 16 lb. per ton weight? 

7. A 7-h.p. engine is used to raise a weight of 24 tons. How 
high will it raise it in 1 min.? If it is desired to raise the weight 
in twice the time, what horsepower is required? 



WORK AND POWER 93 

8. It is desired to raise a weight of 5 tons 200 ft. in 40 min. 
What horsepower is necessary to do the work ? 

9. What average horsepower is required merely to lift a 250- 
ton train up 5 miles of grade rising 1 ft. in every 528 ft., if the 
train is moving at the rate of 25 miles per hr.? 

10. If the name plate on a motor gives the motor rating as 5 
h.p., how many foot-pounds of work should the motor do per min. 
without overheating? Can the motor be worked above its rated 

horsepower? 

PLAN 

11. From the rule that horsepower per cylinder = -^^ in 

which P is the average steam pressure in pounds per sq. in., L the 
length of stroke in ft., A the area of the piston in sq. in., and A^ the 
number of strokes per sec, find the horsepower developed on both 
sides of a locomotive whose cylinders are 20 in. X 28 in., working 
with an average steam pressure of 180 lb. per stroke and 150 
strokes per min.? 

12. A 12-ton casting is raised 20 ft. in 1 min. by a traveling 
crane. What horsepower does the motor on the crane exert? 

13. From the rule for horsepower per cylinder, h.p. = ^^ ^^^ in 

which P is the average pressure in pounds per sq. in. on the piston, 
L the length of stroke in feet, A the area of piston in square inches, 
and A^ the strokes per min. (or 2 times the r.p.m.) find the horse- 
power for both cylinders of the following locomotive. Cylinders 
24 in. X 26 in. (24 in. diam. of piston and 26 in. stroke of piston), 
average steam pressure 160 lb. per sq. in., and running at 200 r.p.m. 

14. From the rule in the foregoing problem calculate the total 
cylinder horsepower of a 4-cylinder Mallet locomotive. 

Cylinders 27 in. X 28 in., average steam pressure on piston 
130 lb. per sq. in., and r.p.m. 150. 

15. Find the horsepower of a freight locomotive running at 
10 m.p.h. with an average cylinder pressure of % boiler pressure. 
Drivers 62 in. in diam., cylinders 24 in. X 28 in., boiler pressure 
205 lb. per sq. in.. 

The power delivered by an engine or a motor at its 
pulley, that is, the actual power available for doing work 
may be found for small machines by the Prony- Brake 
Method. 



94 MECHANICS AND ALLIED SUBJECTS 

Fig. 98 shows such a brake consisting of shoes s and si, 
which are clamped to the machine pulley by bolts b 
and bj which may be regulated to increase or decrease 
the pressure of the shoes on the pulley. When the pulley 
rotates in the direction of the arrow, the entire brake and 
lever L tend to rotate in the same direction. This 
tendency is counterbalanced by the pull on the spring 
shown, from which we may read the pounds pull for dif- 
ferent degrees of pressure of brake shoes on pulley. 

The spring keeps the lever 
^ arm practically horizontal 

'Ey^ throughout the test. The 

power absorbed by the 
->— — -"^ '^5.^'^ biake shoes ''s'' and "si'' 

Jnlf^\li ^Q^^ls ^^^ amount of work 

[yw/n done by the revolving pul- 

^ > , | ley on the machine. In 

•pj^ gg using the brake we first 

take the reading of the 
spring balance with the machine not running in order 
to account for the weight of the lever itself, then after 
bringing the machine up to speed and putting a ^^load^' 
on it by means of bolts b and b, the balance reading is 
again taken, and the initial reading subtracted to get the 
net pull due to the rotation alone. If P is the pull in 
pounds on the balance, L the length of the lever as 
shown andr.p.m. the revolutions per min. of the pulley, 
the foot-pounds of work done per min. equals P X 2tL X 
r.p.m., and the horsepower equals 

P X 27rL X r.p.m. 
33,000 
When used in tests of this kind the machine pulley 
is made so that water may be run on the inside of the 



WORK AND POWER 95 

rim to keep the pulley cool and absorb the heat produced 

by the transformation of the mechanical work done at the 

pulley. 

The principle of the brake described above is the same 

as that of the power of a belt in which the effective belt 

pull is ^'P'' pounds on a pulley of L radius and running at 

a speed represented by the symbol r.p.m. In this case 

the belt moves at a speed of 27rL X r.p.m. feet per min. 

, ,, , • .u ^ P ^ ^ttL X r.p.m. 
and the horsepower is therefore qq ann ^^ 

the same rule as derived above. 

The principle of the brake may also 
be considered the same as that of a 
weight ^^P'' being raised by a rope 
wound on a pulley of radius L, as in 
Fig. 99. 

Example, — Find the horsepower de- 
livered by a direct-current motor in a 
brake test in which the net effective 
pull at end of lever 23^^ ft. long is 
10 lb., with the pulley rotating at 900 r.p.m. 

^ , ,. „ PX27rL X r.p.m. 
Solution, — Horsepower = ^^ ^^^ 

10 X 2 X 3.14 X 2.5 X 900 
^^ 33,000 

which equals 4.282 or practically 4.28 h.p. 

PROBLEMS 

16. Find the horsepower of an engine whose shaft rotating at 
90 r.p.m. produces a pull of 1800 lb. at the end of a brake arm 9 ft. 
long. 

17. In a brake test on a motor it was found that with a speed 




96 MECHANICS AND ALLIED SUBJECTS 

of 750 r.p.m., a pull of 20 lb. was produced at the end of a 5-ft. 
lever. What horsepower did the motor deliver? 

To find the pull produced at the end of a given lever at a given 
speed and horsepower we have by transposing the horsepower rule, 

_ horsepower X 33,000 
" 27rL X r.p.m. 

Writing this equation for the length of lever arm ''L" we have 

_ horsBpower X 33,000 
P X 27r X r.p.m. 

and for the revolutions permin., we have 

^ , ,. . horsepower X 33,000 
Revolutions per mm. = 5 ^y or 

These rules enable us to find any one of the variable quantities 
in the rule in terms of the other quantities. 

PROBLEMS 

18. What pull ''P'' is produced on the end of a 4-ft. brake lever 
when the machine pulley is making 250 r.p.m. and machine is 
delivering 8 h.p.? 

19. Find the brake horsepower of a steam engine running at 
250 r.p.m. and exerting a pull of 80 lb. at the end of a lever 6 ft. 
long? 

20. What length of lever arm is required for a pull of 18 lb., 
with a speed of 180 r.p.m. and a load of 2 h.p.? 

The brake horsepower is the power of an engine or 
motor or water wheel available for use at the machine 
pulley and does not include the power lost in the machine 
itself through friction. 

If in a steam engine we found the horsepower developed 
in the cylinder and called the cylinder or indicated horse- 
power we would find that this power is greater than the 
brake horsepower by the amount of power lost in friction 
in the engine mechanism between the cylinder and the 
fly-wheel or pulley. The horsepower thus lost is called 
the frictional horsepower. 



WORK AND POWER 



97 



In finding the indicated or cylinder horsepower of an 
engine we use a steam indicator or as it may be called, 
a recording steam gauge which takes a record of the steam 
pressure in the cylinder throughout the stroke of the 
piston. Such a card diagram or record is shown in 
Fig. 100. In making a test indicator cards are taken 
at each end of the cylinder. From these we find the 
average pressure throughout the stroke by finding the 
area of the card, dividing this by the length of the card, 
and multiplying by the constant of the indicator. For 




Fig. 100. 




example, if the card shown has an area of 2.10 sq. in. 
and its length is 3.73 in., the average height = 



2.10 
3.73 



.563 in. If the scale of the diagram is 80 lb. to the 
inch this represents an average pressure of .563 X 80 or 
45 lb. per sq. in. 

From this average pressure called the mean effective 
pressure we calculate the horsepower from the rule 



Horsepower = 



PLAN 



33,000 



In which P is the mean effective pressure. 
L is the length of stroke in ft. 
A is the area of the piston in sq. in. 
N is the number of strokes per min. 



98 



MECHANICS AND ALLIED SUBJECTS 




"OLD IRONSIDES" 
1832. Sirflle pair of driving wheeit 
Weight aboyt 6 tons 






CONSOLIDATION TYPE 

Built in 1876. Four pairs of driving wheeli 
Weight in v^orklng order 80 tons 




SANTA FE TYPE 

Built in 1903. Five pairs of driving wheels 

Weight in workinp order 225 tons 




TRIPLEX LOCOMOTIVE 
tin 1914. Twelve pair* of drMng wh«eii 
Weight In working order 42S torn 



Fig. 102. — Cut showing the development in size and weight of 

locomotives. 



WORK AND POWER 99 

To find the area of the indicator card shown, in Fig. 
100 we make use of a planimeter or plane meter as shown in 
Fig. 101. This consists of the two arms A and Ai 
hinged at H and with pin points at P and Pi. The point 
Pi is fixed and the point P is moved around the outhne 
of the card as shown, while the wheel W moves and 
carries a scale which indicates directly the number of 
square inches in the area traced around when the point 
P has completed the outline. 



CHAPTER XIV 
CALCULATION OF BELTING. ENERGY 

38. Horsepower of Belting. — When two shafts are too 
far apart to connect them by gears for transmitting 
power, belts are used. The power which a belt trans- 
mits depends on the effective pull tending to turn the 
driven pulley and the speed with which the belt is 
traveling. The pull which a belt can safely stand 
depends on its material, width and thickness. They 
are sometimes made of one, two, three or four thicknesses 
of leather and are called single, double, triple or quad- 
ruple. The widths of belts vary from about an inch to 
2 ft. depending on the amount of power transmitted. 
Whether single, double or other thickness of belting is 
used depends largely on the size of pulleys on which the 
belt runs. Practice shows that single belts are best on 
pulleys less than a foot in diameter, and double belts on 
pulleys a foot or larger in diameter. On main drives or 
in cases where the amount of power transmitted would 
cause an excessively wide double belt a triple or quad- 
ruple belt is used. 

It has been found that single leather belts will operate 
satisfactorily with an allowable pull of about 30 lb. per 
in. of width. For a double belt about 65 or 70 lb . is 
allowed per in. of width. Almost any mechanical 
handbook contains figures taken from the results of 
practice for the best allowable sizes of belts. 

100 



CALCULATION OF BELTING 101 

Practice shows that the Hnear speed of belts should 
not exceed 4500 ft. per min. if the belt is to make good 
contact with the pulleys. As usually operated, speeds 
below this value are used. 

If P is the allowable pull per in. of width of belt, 
W the width of belt in in., the pull times the width, 
that is P X TF, gives the total effective pull or the force 
transmitted by the belt. This force in lb. multiplied 
by the speed of the belt in ft. per min. gives the foot- 
pounds of work per min. transmitted by the belt. 

To obtain horsepower we divide foot-pounds per min. 
by 33,000, since 1 h.p. is the performance of 33,000 ft.- 
Ib. of work per min. If S is the speed of the belt in ft. 
per min. and using the letter given above for the effec- 
tive pull on the belt, we have the rule 

PXW XS 



Horsepower transmitted by belt = 



33,000 



Example, — Find the horsepower transmitted by a belt 
5 in. in width, if the allowable pull per in. of width is 
35 lb. and the speed of the belt 2400 ft. per min. 

Solution: 

^ PXW X S 35X5X 2400 
Horsepower = —^^^^^^^- or 33-^-^^^ 

which equals 12.7. 

To find the width of belt to transmit a given horse- 
power for a given belt speed, we transform the above 
equation to express W in terms of the other quantities 
involved as follows. 

If H represents the horsepower transmitted we have as 

, ^ PXW XS ,^, . ^,, HX 33,000 

above, H = — ^^ — and therefore W = p k/ q — * 

Example. — Find the width of double leather belt re- 



102 MECHANICS AND ALLIED SUBJECTS 

quired to transmit 100 h p., if the speed of belt is 4000 ft. 
per min. and the allowable pull per in. of width is taken 
as 70 lb. since this is a double belt. 
Solution: 

^,, H X 33,000 100 X 33,000 , . , i ., o 

W = —pT^g — or 7Q y^ 400Q which equals 11.8 

in. Therefore a 12-in. belt should be used. 

PROBLEMS 

1. Find the width of a single belt required to transmit 95 h. p. 
when belt runs on a 14 ft. diam. pulley at 70 r.p.m. when P = 40 lb. 

Note. — The speed of the belt in ft. per min. equals the product 
of the circumference of the pulley in ft. by the r.p.m. of the 
pulley. 

2. Find the width of belt required to transmit 15 h.p., if belt 
speed is 3500 f .p.m. and allowable pull per in. of width of belt is 
501b. 

39. Definitions on Energy, and Problems. — Energy is 
power to do work. 

All objects possess energy on account of work having 
been done upon them at some time. Energy, like work, 
is measured in foot-pounds. There are, in general, two 
kinds of energy due either to the motion or to the position 
of objects. Energy of motion is called kinetic energy 
and energy of position is potential energy. For example, 
an object set in motion can overcome a certain amount of 
resistance before being brought to rest, and the energy 
which the object has on account of its motion is used up 
in overcoming the resistance, bringing the object to rest. 
Fly-wheels on engines both receive and give up energy 
and thus cause the engine to run more smoothly through- 
out the stroke. 

Elevated weights have power to do work on account 



CALCULATION OF BELTING 103 

of their elevated position, as in various types of hammers, 
pile drivers, etc. The rule for finding the kinetic energy 
{K,E.) of an object is: 

In which W is the weight of the object in lb., v is 
its velocity or speed in ft. per sec. Thus a weight of 
230 lb. moving with a speed of 1200 ft. per sec. has a 

, . ^. ^ 230 X 1200 X 1200 . ^ , 

kinetic energy of oaTa ^^ approximately 

5, 140,000 ft.-lb. The rule for potential energy {P,E.) is: 

P,E, = W X H 

W is the weight of the object in lb. and H is its 
height in ft. above a given point. The energy is then 
given in foot-pounds. For example: A 2-ton steam 
hammer when raised 4 ft. has a potential energy of 
2 X 2000 X 4 or 16,000 ft.-lb. on account of its posi- 
tion and if let fall it would do 16,000 ft.-lbs. of work. 

PROBLEMS 

3. A weight of ^^ ton is moving with a speed or velocity of 
5 ft. per sec. How much energy in foot-pounds is stored up in it? 

4. The rim of a fly-wheel weighing 3 tons moves with a speed 
of 20 ft. per sec. How many foot-pounds of energy are stored, 
up in it? 

5. A 3-ton hammer is raised and dropped a distance of 8 ft., 
8 times per min. How much work does it do per min.? 

6. Three thousand gallons of water are pumped into a tank 
an average height of 10 ft. above the level of a track. What energy 
does this body of water have with respect to the track? 

7. If a punch makes a hole through a 3^2-in. steel plate and 
meets with an average resistance of 30 tons, how much energy is 
consumed in the punching alone ? 

8. A machinist wields a 14-lb. hammer and strikes withaveloc- 



104 MECHANICS AND ALLIED SUBJECTS 

ity of 30 ft. per sec. and 28 blows per min. How many foot- 
pounds of energy are produced per min.? At what horsepower 
does he work? 

9. How much work in foot-pounds is required to raise a 10- 
ton casting vertically through a height of 5 yd.? 

10. If 10,000 cu. ft. of water flow per min, over a fall 20 ft. high, 
how many foot-pounds of energy does this water deliver at the 
foot of the fall? (1 cu. ft. of water weighs 62.5 lb.) 



CHAPTER XV 
HEAT 

40. Definitions. — Heat is one form of energy. Other 
forms of energy such as electrical, chemical, energy of 
position and of motion can be changed into heat. Heat 
is produced by friction, by chemical action, and by the 
electric current flowing through a resistance such as 
copper or iron wire. In lighting a match heat is pro- 
duced both by friction, and by chemical action in the 
burning of the match. The heat produced by a cutting 
tool on the metal being turned off in a lathe, and also 
that produced in grinding tools, are familiar examples 
of heat produced by friction, or in other words, by the 
change of mechanical work into heat. The burning of 
fuel is an example of a chemical change involving heat 
action. Objects in motion become heated when brought 
to rest by friction or by striking other objects. 

41. Temperature. — The temperature of an object 
means not only whether the object is ^^hof or ^^cold,^' 
but whether it takes heat from, or gives heat to, sur- 
rounding objects. When two objects at different tem- 
peratures are brought together there is a tendency toward 
an equalization of temperature. Like water which flows 
from a point of high to one of low level, heat flows from 
an object of high to one of low temperature. For meas- 
uring temperature, in most cases for ordinary work, 

105 



106 



MECHANICS AND ALLIED SUBJECTS 



200 
ISO- 
ISO— 
170— 
160— 
150 — 
140— 
130— 
120— 
110 — 



212 F= 100 C 



BOILINQ 



mercury thermometers are used, either the Fahrenheit 

or the Centigrade as shown in Fig. 103. 

The Centigrade thermometer has 100 equal divisions 

on its scale between the point of melting ice and boiling 

water, with the mark at the 
point of melting ice. 

The Fahrenheit thermometer 
has 180 equal divisions on its 
scale between the points where 
ice melts and water boils. The 
32- degree mark is placed at the 
point where ice melts and the 
212-degree mark at the point 
where water boils. 

To change from a temperature 
in degrees Fahrenheit to the cor- 
responding temperature in de- 
grees Centigrade, and vice versa, 
the following rules are used. 

9 



100— 
80— 
SO- 
TO— 
60- 
BO— 
40 — 



82 F- 



20— 
10— 
0— 

-10— 
-20— 



-40- 



32 F« C 



FREEZING 



100 



90 



80 



— 70 



60 



— 50 



40 



— 20 



— 10 



F.° = C. 



X 5 + 32. 







-10 



C.° = (F.° - 32) 



9 



30 



—40 



For example, using the first 
rule, suppose we wish to find the 
temperature Fahrenheit corre- 
sponding to 30° Cent., we have 

F.° = 30° X ^ + 32 

or F.° = 54 + 32 or 86°. 

That is, 86° on the Fahrenheit thermometer correspond 
to 30° on the Centigrade thermometer. 



i m 



Fig. 103. 



HEAT 107 

Working the other way round and using the second rule 

30 = (F. -32)|or30 = ^-^ 
or 270 = 5 F. - 160 and F. = 86^ 

Alcohol and air thermometers are also used for meas- 
uring temperatures and these as well as the mercury 
thermometers make use of the principle of expansion 
and contraction of the alcohol, air or mercury with differ- 
ent temperatures, to obtain different readings. 

42. Heat Measurement. — Heat is measured in British 
Thermal Units (abbreviated B.t.u.). One B.t.u. is the 
amount of heat required to raise the temperature of 
1 lb. of water 1 ° Fahr. 

For example, if 10 lb., of water are heated from 60"^ 
Fahr. to 90° Fahr. the amount of heat taken up by 
the water equals 10 X (90° - 60°) or 10 X 30 or 300 
B.t.u. 

When water reduces in temperature it of course gives 
up heat and the calculations are made in the same way 
by multiplying the weight of water in lb. by the 
drop in temperature in degrees Fahrenheit. This gives 
the number of B.t.u. liberated by the water. It should 
be understood that temperature means heat '^intensity'' 
or as we might say heat ^^ pressure'^ and is measured in 
'^degrees,'' while ^^ quantity'' of heat is measured in 
B.t.u. We can make the same comparison with air, in 
which the ^^ntensity'' or ^^ pressure'' is measured in ^'Ib." 
by a gauge, while the ^'quantity" of air is measured in cu. 
ft. Again in the case of steam, we measure the ^^ pres- 
sure" in ''lb." and the ''quantity" in lb. of condensed 
steam. 

If two iron castings are of different weights, it will take 



108 MECHANICS AND ALLIED SUBJECTS 

a different ^^ quantity ^^ of heat in B.t.u. to heat them to 
the same ^4ntensity'' or ^temperature/' 

In the same way two air cyUnders can contain different 
quantities of air and still both be at the same pressure. 

If the change of temperature is given in Centigrade 
rather than in Fahrenheit degrees we multiply the 
number of Centigrade degrees by % since 1 Centigrade 
degree equals % Fahrenheit degrees. 

If the weight is given in units other than pounds we 
first change the weight to the equivalent number of 
pounds. 

Example. — How much heat in B.t.u. is taken up by 

1 ton of water which is heated from 10° Cent, to 40° 

Cent.? 

9 
Solution,--B.t,u. = 1 X 2000 X (40-10) X ^ or 2000 

30 X 9 
X — r — which equals 108,000. Arts, 

43. Expansion and Contraction Due to Heat. — Most 
substances expand or increase in volume when heated 
and contract or decrease in volume when heat is removed 
from them. 

This effect takes place in solids, liquids, and gases. 
The principle of expansion and contraction is made of 
considerable use and in some cases it may cause consider- 
able trouble unless allowed for. 

When a piece of metal, for example, is heated, it ex- 
pands in all directions. The expansion in one direction 
or dimension is that with which we are particularly 
concerned and is called the ^^ linear'^ expansion of the 
substance. 

For example, in the case of mercury thermometers the 
principle of expansion is used, also in steam piping, 



HEAT 109 

allowance must be made for the lengthening and short- 
ening of the pipe system due to heat changes. In laying 
steel rails an allowance is made between the ends of the 
rails for contraction and expansion. The same is true 
in the case of bridges and other metal structures. 

The power of expansion and contraction of metal is 
very great and if allowance is not made for it the expan- 
sion will take place just the same with the result that the 
part expanding will buckle or give itself the necessary 
room for expansion in one way or another. These char- 
acteristics of metals are made use of in putting tires 
on wheels, in making shrink fits and numerous other 
applications. 

In the case of tires put on wheels and shrink fits of 
various kinds, an allowance is made per in. of diameter 
of the object on which the fit is to be made of approxi- 
mately Kooo of an in. and in the case of a locomotive 
driver 68 in. in diam. the tire would be turned 68 times 
Hooo or ^%ooo of an in. smaller in diam. than the center 
on which it is to go. It is then heated so that it will 
expand sufficiently to slip on to the wheel before it cools. 
As it cools it contracts and grips the wheel. As it cannot 
return entirely to its original size this gripping power is 
sufficient to hold the tire in place during severe service. 
In case of a crank to be fitted on a 6-in. shaft the crank 
should be turned about %ooo of an in. smaller than the 
shaft or else the shaft made %ooo of an in. too large and 
the crank made just 6 in. in diam. This, as explained 
will make the proper allowance so that the crank will 
grip the shaft tightly after it has been shrunk on. 

The amount of expansion of an object depends upon 
the kind of material, upon its length and upon the change 
in temperature of the object. Tables are prepared, which 



no MECHANICS AND ALLIED SUBJECTS 

give us for different materials the amount of expansion 
per unit length, per degree change in temperature. 
These figures are called ^^coefficients of linear expansion/' 
The following figures for some of the most used mate- 
rials are sufficiently accurate for our purposes. 

Coefficients of Expansion of Solids 

Tv/r^i.„i Coefficient 

Aluminum (cast) 00001234 

Brass, cast 00000957 

Brass, plate 00001052 

Concrete: cement, mortar and pebbles. . . .00000795 

Copper 00000887 

Iron, wrought 00000648 

Iron, cast 00000556 

Steel, cast 00000636 

Steel, tempered 00000689 

The coefficient for 1 Centigrade degree of temperature 
may be obtained from those above by multiplying them 
by%. - _ 

To calculate the amount of expansion the rule is, multi- 
ply the coefficient of expansion by the length of the object 
and by the number of degrees rise in temperature. If 
the length is taken in ft. the expansion will be in ft. 
If the length is in in. the answer will be in in., etc. 

Example. — What is the amount of expansion of an 
18-ft. brass rod when heated 150° Fahr.? 

Solution. — Amount of expansion = Coefficient of ex- 
pansion X length of rod X degrees rise in temperature. 

Expansion = .0000096 X 18 X 150 or .0259 ft. 

Example. — What will be the increase in diam. of a 

64-in. diam. locomotive driver when heated 500° Fahr.? 

Solution. — The coefficient of expansion of steel is taken 



HEAT 111 

as .0000064. The expansion therefore equals .0000064 X 
64 X 500 or .2048 in. 

Heat also causes liquids and gases to expand, but in 
the case of gases the change of volume depends upon 
the pressure of the gas as well as upon its temperature. 
Solids are liquefied by heat as in melting iron and steel 
and liquids are changed into vapors as in the production 
of steam in the locomotive boiler. 

PROBLEMS 

1. Find the expansion in an aluminum rod 20 ft. long if it is 
heated 750° Fahr. 

2. What is the approximate expansion in the length of a steel 
bridge 2800 ft. long through a difference of temperature of 0° 
Fahr. in winter and 90° Fahr. in summer? 

3. The walls of a building 40 ft. wide were drawn in and straight- 
ened by using iron rods through them, placing plates on the ends 
of the rods, heating the rods, tightening up the plates by means 
of nuts and allowing the rods to cool and contract, thus drawing 
the building together. If the rods each time changed in tempera- 
ture 800° Fahr. how much was the building drawn together with 
each heating and tightening of the plates? 

4. What is the amount of expansion in a 200-ft. span of copper 
wire which undergoes a change of temperature of 80° Fahr. ? 

5. An iron steam pipe changes in temperature from 180° Fahr. 
to 8° Fahr. What allowance must be made for the change in 
length if the pipe is 160 ft. long at 180° Fahr.? 

6. If steel girders on a bridge are 60 ft. long at 20° Fahr., what 
is their length at 94° Fahr. 

7. What allowance must be made for a crank to be fitted to a 
5-in. diam. shaft? 

8. Find the expansion in a 40-ft. rail between the temperature 
of - 20° Fahr. and 90° Fahr. 

44. Specific Heat. — The specific heat of a substance 
means the number of heat units (B.t.u.) required to 
raise the temperature of any weight of the substance 



112 MECHANICS AND ALLIED SUBJECTS 

1°, as compared with the number of heat units (B.t.u.) 
required to raise the temperature of the same weight of 
water 1°. 

The following are examples of specific heat: Steam 
.48, iron .114, lead .0314. 

This means that to raise the temperature of 1 lb. of 
iron 1° Fahr. requires only .114 B.t.u., whereas to raise 
the temperature of 1 lb. of water 1° Fahr. requires 1.0 
(one) B.t.u. 

45. Latent Heat. — Latent or '' hidden heaf is that 
required to change the condition of a substance from a 
solid to a liquid or from a liquid to a vapor without 
changing its temperature. 

To vaporize 1 lb. of water at 212° Fahr. to steam at 
212° Fahr. at ordinary pressure requires 966 B.t.u. To 
melt 1 lb. of ice at 32° Fahr. to water at 32° Fahr. requires 
142 B.t.u. of heat. 

46. Mechanical Equivalent of Heat. — The mechanical 
equivalent of heat means the relation between units of 
work in foot-pounds, and equivalent heat units. We all 
know that friction produces heat and to overcome the 
friction work must be done. When 778 ft. -lb. of work 
are done, an amount of heat is produced equal to IB.t.u. 
and this amount of heat is sufficient to raise the tem- 
perature of 1 lb. of water 1° Fahr. 

Example, — How much heat is produced by 10,000 
ft. -lb. of work? 

Solution: 10,000 .oc-pf a o 

rj„^ or 12.8 B.t.u. Ans, 

47. Heat Produced by the Electric Current. — The 
heat in B.t.u. produced by an electric current flowing 
through a resistance is equal to the square of the cur- 



HEAT 113 

rent in amperes times the resistance of the circuit in 
ohms times the time in sec. during which the current 
flows and times .00095 or 

Heat in B.t.u. = (the current in amperes)^ X resistance 
in ohms X seconds X .00095 

Example, — How much heat is produced by a current 
of 10 amperes flowing for 15 min. through a resistance 
of 22 ohms? 

B.t.u. = 10 X 10 X 22 X 15 X 60 X .00095 or 1881. 

Arts. 

Working the other way round we can find the current 
required to produce a given amount of heat when flowing 
for a given time through a given resistance. 

If H is the heat in B.t.u., I the current in amperes, 
R the resistance in ohms, and T the time in sec, the rule 
given above may be written, 

H = PXRXTX .00095. 

To find the current ^^/^' in terms of the other quantities 
we have 

p = -H 

RXT X .00095 



and therefore I = \ h^ 1^ ^^^^^ 

\RXTX .00095 

Example. — What current in amperes is required to 
produce 5000 B.t.u. in 10 min. when flowing through 
a resistance of 40 ohms? 

Solution: 



5000 / 5000 

\'40 X 10 X 60 X .00095 ^^ \24,000 X .00095 



This equals ^/ or 14.8 amperes. 



114 MECHANICS AND ALLIED SUBJECTS 

To find ^^R'^ from the above rule in terms of the other 
quantities we have 

H 



R = 

Similarly T = 



PX T X. 00095 
H 

PX R X .00095 



PROBLEMS 



9. A Centigrade thermometer in a boiler room reads 40° 
what is the corresponding temperature Fahrenheit? 

10. The difference between the temperature of two objects is 
20 Fahr. degrees. Express this difference in Cent, degrees. 

11. What effect would there be in the readings of a mercury 
thermometer if the tube were not of uniform bore? 

12. A Fahrenheit thermometer gives the temperature of water 
as 95^. What is the corresponding temperature in degrees 
Cent.? 

13. How much heat is required to raise 100 lb. of water 10"* 
Fahr.? 

14. If 10 cu. ft. of water change in temperature from 90° Fahr. 
to 75° Fahr., how many heat units are liberated? 

15. It is desired to raise 5000 lb. of water from 60° Fahr. to 
90° Fahr. and supposing 10 per cent, of the heat value of the coal 
to be used, how much coal is required if there are 12,000 B.t.u. 
per lb.? 

16. How much heat is required to raise 100 lb. of iron from 70° 
Fahr. to 350° Fahr.? To how many foot-pounds of energy does 
this correspond? 

17. Which requires more energy to raise the temperature of 10 
lb. of iron from 0° to 100° Fahr. or to lift 5 tons of iron 1 ft. verti- 
cally? 

18. If 1 lb. of coal contains 12,000 B.t.u. to what height could 
a 1200-ton weight be raised by utilizing all the heat value of the 
coal? 

19. An engine raises a casting weighing IJ2 tons through a 
vertical distance of 5 ft. What is the number of B.t.u. which 
this amount of work represents? 



HEAT 115 

20. How many B.t.u. are required to raise 50 lb. of lead from 
72"Fahr. to 520" Fahr.? 

21. A car weighing 15 tons is lifted by a crane through a vertical 
distance of 10 ft. Find the number of foot-pounds of work done 
and the number of B.t.u. which the work represents. 

22. A certain coal gives 14,000 B.t.u. per lb. To what height 
would 5 lb. of this coal lift a weight of 8 tons if 40 per cent, of the 
B.t.u. were utilized? 

23. If 2 tons of water are used per hr. in an ice plant in freezing 
water from an average of 70° Fahr., how many B.t.u. are extracted 
from this water per day of 10 hr.? 

24. Which requires the more energy, to raise the temperature 
of 20 lb. of lead 100° Fahr. or to lift 4 tons of iron 12 ft. vertically? 

25. If 1 lb. of coal contains 12,000 B.t.u. to what height could 
a 10-ton weight be lifted, if 5 per cent, of the heat value of the coal 
were utilized? 

26. A wire cage weighing IJ^ tons is lifted 900 ft. four times an 
hr. How many foot-pounds of work are done on the cage per hr. ? 

27. If 3 per cent, of the heat value of a 12,000 B.t.u. coal were 
utilized, how many pounds of coal would be used per hr. in lifting 
the cage in Prob. 26? 

28. Find the heat produced by a current of 20 amperes flowing 
for 30 min. through a resistance of 24 ohms. 

29. How long will it take a current of 10 amperes to produce 
6000 B.t.u. when flowing through a resistance of 60 ohms? 

30. What resistance is necessary in order to develop 4800 
B.t.u. in 30 min. with a current of 8 amperes? 



CHAPTER XVI 
COMMON OR BRIGG'S LOGARITHMS 

48. Definitions. — It is a good thing to become familiar 
with the use of logarithmic tables and common rules for 
using the logarithms of numbers. 

In working out problems with logarithms we make 
use of a table of logarithms from which we can get the 
logarithm of any number of three, four or five figures, 
depending on how many figures the tables are made to 
give. 

A logarithm of a number is expressed in two parts, one 
called the characteristic and the other the mantissa. The 
characteristic is obtained directly from the number and 
in the logarithm is the figure to the left of the decimal 
point. The mantissa is found in the tables and is the 
figure to the right of the decimal point in the logarithm. 

49. The Characteristic. — When the number whose 
logarithm w^e are finding is greater than one the character- 
istic is positive, or plus. When the number is less than 
one the characteristic is negative j or minus. The charac- 
teristic is determined froni the number of figures which 
make up the number whose logarithm we are finding, 
and is always one less than the total number of figures to 
the left of the decimal point. For instance, the character- 
istic of the logarithm for the number 3354 is 3 because 
there are four figures in the number. The characteristic 
of the logarithm for the number 384 is 2, for 24.43 it is 1, 
and for 7 is 0. There are all positive characteristics as 
the numbers were all greater than 1. 

116 



COMMON OR BRIGG'S LOGARITHMS 
Table 1.^ — Common Logarithms 



117 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0756 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


17 


2304 


2330 


2355 


2381 


2405 


2430 


2455 


2480 


2504 


2529 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


32 


5051 


5065 


5079 


5092 


5105 


5119 


5132 


5145 


5159 


5172 


33 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


34 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


35 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 


38 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 


5888 


5899 


39 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6222 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


44 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


46 


6628 


6637 


6646 


6636 


6665 


6675 


6684 


6693 


6702 


6712 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6785 


6794 


6803 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 


6875 


6884 


6893 


49 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 



118 MECHANICS AND ALLIED SUBJECTS 

Table 1. — Common Logarithms. — Continued 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 


7474 


56 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7536 


7543 


7551 


57 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 


58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 


59 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7760 


7767 


7774 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


61 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 


63 


7924 


7931 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 


63 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 


64 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


66 


8195 


8202 


8209 


8215 


8222 


8228 


8235 


8241 


8248 


8254 


67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 


68 


8325 


8331 


8338 


8344 


8351 


8357 


8363 


8370 


8376 


8382 


69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


8445 


70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


S494 


8500 


8506 


71 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8561 


8567 


73 


8573 


8579 


8585 


8591 


8597 


8603 


8609 


8615 


8621 


8627 


73 


8633 


8639 


8645 


8651 


8657 


8663 


8669 


8675 


8681 


8686 


74 


8692 


8698 


8704 


&710 


8716 


8722 


8727 


8733 


8739 


8745 


75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 


77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8910 


8915 


78 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8960 


8965 


8971 


79 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 


80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 


83 


9138 


9143 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 


83 


9191 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


84 


9243 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 


85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


86 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 


87 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 


88 


9445 


9450 


9455 


9460 


9465 


9469 


9474 


9479 


9484 


9489 


89 


9494 


9499 


9504 


9509 


9513 


9518 


9523 


9528 


9533 


9538 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 


9586 


91 


9590 


9595 


9600 


9605 


9609 


9614 


9619 


9624 


9628 


9633 


93 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


9675 


9680 


93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


9717 


9722 


9727 


94 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 


95 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 


97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


98 


9912 


9917 


9921 


9926 


9930 


9934 


9939 


9943 


9948 


9952 


99 


9956 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 



COMMON OR BRIGG'S LOGARITHMS 119 

The negative characteristics for numbers less than one 
are determined in the same manner but they are expressed 
differently. Thus, the characteristic of the logarithm for 
the number .562 is — 1 (minus one) and instead of writing 

— 1 for the characteristic it is customary to change its 
form by adding 10, or a multiple of 10 to the character- 
istic, and then indicating the subtraction of the same 
number from this result. Thus a —1 characteristic is 
written 9. (mantissa) -10. For -10 + 9 = -1. The 
characteristic for the logarithm of the number .062 is 

— 2 and is written 8. (mantissa) —10, since —10 + 8 = 
-2. 

50. The Mantissa. — The mantissa consists of the 
figures obtained from the tables and comprises the 
portion of the logarithm to the right of the decimal 
point or the decimal part of the logarithm and is always 
found by the use of the tables. 

The total logarithm for a number is obtained by com- 
bining the mantissa read in the tables, with the proper 
characteristic. 

51. Use of the Tables. — The tables give the mantissa 
of the number only, and are arranged to read the first 
two figures of the number of which we wish to find the 
logarithm under the column headed N. and the third 
figure of the number to the right of N. at the top of the 
table. For example, if we wish to find the logarithm of 
the number 215, look down the vertical column under 
N. for the first two figures of the number or 21 and then 
across the page horizontally to the vertical column headed 
with the third figure of the number or 5 and read the 
figures 3324 which is the mantissa of the logarithm for 
the number 215. 



120 MECHANICS AND ALLIED SUBJECTS 

The characteristic for 215 is 2. Hence the logarithm 
of 215 is 2.3324. 

The logarithm of the number 384 is 2.5843. (2 is the 
characteristic) (.5843 is the mantissa.) 

The logarithm of the number .287 is— 1.4579 ( — 1 is 
the characteristic) (.4579 is the mantissa). 

The logarithm of the number 26 is 1.4150. (1 is the 
characteristic and .4150 is the mantissa.) 

The logarithm of the number .236 is 9.3729 -10. 

— 1 or (9 — 10) is the characteristic and .3729 is the 
mantissa. 

The logarithm of the number .0867 is 8.9380 - 10. 

— 2 or (8 — 10) is the characteristic and .9380 is the 
mantissa. 

The logarithm of 8.3 is 0.9191. is the characteristic 
and 0.9191 is the mantissa. 

52. To Find a Number Corresponding to a Logarithm. 
— We find the nearest logarithm to the one given and 
look across horizontally under column N. for the first 
two figures of the number and to the top of the column 
in which the logarithm occurs for the third figure. 
The characteristic will determine the number of figures 
to the left of the decimal point in the number. Thus: 
Find the number corresponding to the logarithm 2.3366. 
The nearest mantissa is .3365 occurring in the horizontal 
line opposite the number 21 in column headed N. and 
in the vertical column headed 7. Hence the number is 
217. with the decimal point after the seven because the 
characteristic is 2 (the number of figures to the left of 
the decimal point in the number is one more than the 
characteristic). 

There are four (4) general rules for the use of loga- 
rithms. These may be stated as follows: 



COMMON OR BRIGG'S LOGARITHMS 121 

Rule 1. — To multiply two numbers: 

Add their logarithms and find the number in the table 
corresponding to the sum of their logarithms. Thus: 
Multiply 261 by 885 

log 261 = 2.4166 
log 885 = 2.9469 



Sum of logs = 5.3635 

Number = 231000. Arts, 

Pointing off one more figure in the number than the 
characteristic we have six figures before the decimal point 
in the answer. 

PROBLEMS FOR PRACTICE 

1. Multiply 235 by 360. 6. Multiply .279 by 56.3. 

2. Multiply 898 by 210. 7. Multiply .076 by .005. 

3. Multiply 635 by 359. 8. Multiply 9.15 by 12.4. 

4. Multiply 110 by 236. 9. Multiply .028 by .144. 

5. Multiply 2.56 by 304. 10. Multiply 186 by .062. 

Rule 2. — To divide two numbers: 

Subtract the logarithm of the subtrahend from the 
logarithm of the minuend and find the number in the 
table corresponding to the difTerence of these logs. Thus : 

Divide 836 by 2.10 

log 836 = 2.9222 

log 2.10 = 0.3222 
Difference of logs = 2.6000 Number = 398. Arts, 

Since the characteristic is 2.6000 we point off there 
figures in the number, the decimal point being placed 
after the eight. 



122 MECHANICS AND ALLIED SUBJECTS 

EXAMPLES FOR PRACTICE 

11. Divide 600 by 42. 16. Divide 1.04 by .02. 

12. Divide 489 by 5. 17. Divide 98.1 by 12.5. 

13. Divide 628 by 124. 18. Divide .126 by 6.26. 

14. Divide 989 by 2.5. 19. Divide .166 by 1.25. 

15. Divide 9.67 by .002 20. Divide 7.89 by 6.32. 

53. To Raise a Number to a Power. — Multiply the 
logarithm of the number by the exponent or power to 
which the number is to be raised, and find the number 
in the table corresponding to the product thus obtained. 

Exa7nple.— Find the value of (384)2-^ 

Solution: Log 384 = 2.5843. 

2.5 X 2.5843 = 6.4608. The number corresponding to 
the logarithm 6.4608 = 2890000. Ans. The nearest 
number corresponding to the mantissa .4638 is 289. 
The characteristic 6 gives seven figures to the left of the 
decimal point. 

Example. — Find the value of (.0325)^. 

Solution: Log .0325 = 8.5119 - 10 

3 X (8.5119 - 10) = 25.5357 - 30 
or - 5.5357 

From the tables the number nearest 5357 is 343. 
With a characteristic of — 5 the answer is .0000343. 
Example. — Find the value of (.305)'^. 

Solution: Log. .305 = 9.4843 - 10 

7 X (9.4843 - 10) = 66.3901 - 70 
or - 4.3901 

From the tables the number nearest 3901 is 246. 
With a characteristic of — 4 the answer is .000246. 



COMMON OR BRIGG'S LOGARITHMS 123 

PROBLEMS FOR PRACTICE 

Find the value of 

21. (628)= 26. (76.3)02 

22. (988) -2 27. (2.8)2-5 

23. (625)7-8 28. (.002)2-^8 

24. (888)'^^ 29. (6.25)«o2 

25. (.867)2-62 30. (0.0652)2-89 

54. To Find a Root of a Number. 

Divide the logarithm of the number by the index of 
the root and find the number corresponding to the quo- 
tient thus obtained. 

Example. — Find the value of v 286. 

Solution: Log 286 = 2.4564 

?:f* = 0.8188 

From the tables the number nearest .8188 is 659. 

With a characteristic of 0, the answer is 6.59 that is 
we point off one figure in the number thus bringing the 
decimal point after the six. 

Example.— Find the value of ^.000625. 
Solution: Log .000625 = 6.7959 - 10 
Before dividing the logarithm by 6, we add and sub- 
tract from the characteristic a number which is a multiple 
of 10 and which will make the negative portion of the 
characteristic divisible by six without a remainder. If 
we add and subtract 50 the logarithm 6.7959 — 10 be- 
comes 56.7959 - 60. 

Then 5-M9f_^ = 9.46598 - 10 

6 

or 9.4660 - 10 

or -1.4660 

From the tables the number nearest 4660 is 292. 

With a characteristic of — 1 the answer is .292. 



124 MECHANICS AND ALLIED SUBJECTS 

EXAMPLES FOR PRACTICE 

31. \/l02 36. V^265 

32. -^3.85 37. ^4.26 

33. ^^360 38. Vim 

34. 4^ 206 39. ^12.6 

35. V^0265 40. \/32j6 

MISCELLANEOUS PROBLEMS FOR PRACTICE 

(To be solved by logarithms) 

41. Find the circumferences of circles whose diameters are 
as follows: 126 ft. in. 7 ft. 6% in. 6 ft. 11.2 in. 1720 ft. 
in. 7890 ft. in. (Circumference = 3.14^ X D where D = 
diameter.) 

42. What are the areas in sq. in. of circles with the follow- 
ing diameters: 62.2 in. 180 in. 9 ft. 6 in. 7 ft. ll^e in. 5 ft. 
10>f in. (Area = .785Z)2 where D = diameter.) 

43. What are the diameters in in. of circles whose areas are: 
6450 sq. in.? 5010 sq. in.? 4050 sq. in.? 7370 sq. in ? 



'"-Vs 



or D = TT-^-- \/area where D = diameter.) 



785 0.887 , 

44. Find the horsepower of a locomotive from the following 
rule: 

_ PLAN _ 130 X 28X24 X 24 X .785 X 300 X 2 
Horsepower - 33 qqq - 12X33,000 

in which P = 130 lb. per sq. in. pressure on the piston, cylinders 
24 in. X 28 in., N = 300 strokes per min= 

Work out the following problems by logarithms, showing all 
the work: 



45. 



43.8 X 2760 X .97 X 3.14 



7.85 X 2.33 X 42.6 X .785 

46. \/294. V4.84. v^3270. v^8.24. 

47. (29.7)2. (436)3. (2.86)^. (6.78)^ 

48. Find the horsepower of a Pacific type locomotive with cylin- 
ders 24 in. X 26 in., drivers 80 in., running at 40 m.p.h., and ef- 



COMMON OR BRIGG'S LOGARITHMS 125 

fective steam pressure of 160 lb. per sq. in. using the following 

figures : 

PLAN 
Horsepower = oo qqq 

16QX 26 X 24 X 24 X -785 X 40 X 5280 X 12 X 2 X 2 
12 X 33,000 X 80 X 3.14 X 60 

49. An axle weighs originally 1050 lb. It loses 5% per cent, of its 
weight when turned down. Find the weight of the material re- 
moved, the number of cubic inches of material removed, and the 
final weight of the axle. The material w^eighs .28 lb. per cu. in. 

50. Find the capacity in gal. and in cu. ft. of a cylindrical tank 
14 ft. high and 10 ft., in diam. 

51. Design a cylindrical tank to hold 20,000 gal. 

52. Design a coal bin to hold 300 tons of coal (one ton oc- 
cupies about 37 cu. ft.). 

53. How many cylindrically shaped iron castings 4 in. diam. 
and l^i ft. long can be made from 10 tons of metal. (Iron weighs 
.26 lb. per cu. ia.) 



CHAPTER XVII 
THE MEASUREMENT OF RIGHT TRIANGLES 

55. Definition. — In the work which follows are rules 
for finding the sides and angles of right triangles, that is, 
those having a 90 degree (90°) or right angle. These 
rules are used to work out problems in the shop work as 
shown by examples which follow. 

The rules for working out triangles (that is, figures 
with three angles and also three sides) form a branch of 
mathematics called ^ ^ trigonometry, ' ^ ^ ^ tri ^ ^ meaning 




3 = y/c^^^ 
Fig. 104. 



three, ^^gono'' meaning side and ''metry^' measurement, 
or the measurement of three-sided figures. We need not 
think of the work necessarily as ^ trigonometry,^' but 
principally how it apphes in working out problems that 
cannot be done by rules we have had before. We can 
use these rules also to do some of the problems quicker 
than by rules we have had up to this time. 

56. Right Triangle Rule. — We have already had the 
rule that in a right triangle ''the square of the hypothe- 

127 



128 



MECHANICS AND ALLIED SUBJECTS 



nuse is equal to the sum of the squares of the other two 
sides/' that is, referring to Fig. 104, if the hypothenuse 
is Cj the height A, and the base B then 





(hyp.)2 = (height)^ + (base)^ 


that is 


C^ = A'- +B' from this 




C =VA2 + 5" 


also since 


C^ = A^ + B^ 




A^ = C - 52 


and 


A = -s/C - B- 


and also 


B = VC - A^ 



That is when we have any two sides of a right triangle 
we can find the other side from one of these rules. It 
often is necessary, however, to find a side of a right tri- 
angle when we know only one other side and only one 
acute angle of the triangle. For example, in Fig. 105 





B-41.6 
Fig. 106. 



the hypothenuse is 5 in. and the angle a = 30°. If 
we want to find either side A or 5 we cannot use the 
rules A = \^C'^ — B^ and B = VC"^ - A'^ since we know 
only one side in either of these equations. To find either 
side A or jB we use certain ratios between the sides of the 
right triangle. These ratios have already been worked 
out for us in tables for different angles in the triangle. 



MEASUREMENT OF RIGHT TRIANGLES 129 

If as in Fig. 106, angle a = 30°. Side A = 24 and side 

C = 48. 

Side A _ side opposite angle a _ 24 _ 1 __ 
Side C ~ hypothenuse 48 2 "" * 

57. Definition of Sine, Cosine, Tangent, and Cotan- 
gent. — This ratio oi the opposite side and the hypothenuse 
is called the sine of angle a and for 30° it is .5. Similar 
ratios are worked out for all angles between 0° and 90° 
and put into tables for our use. If we divide the ba^e B 
by the hypothenuse we have (adjacent meaning the side 
forming one side of the angle) 

side B _ side adjacent to angle a _ 41.6 
side C hypothenuse 48 ~ ' 

This ratio is called the cosine of angle a. Similar ratios 
are worked out for all angles and put into tables for 
our use. 
If we divide side A by side B we have 



side A side opposite angle a 24 



side B side adjacent to angle a 41.6 

This is called the tangent of angle a. Also 

side B _ side adjacent to angle a _ 41.6 
side A side opposite angle a 24 



,577 



1.73 



This is called the cotangent of angle a. 

These four ratios are what we will use in working out 
our triangles. They are called functions of the angle 
considered. Put in tabular form these functions are: 

9 



130 MECHANICS AND ALLIED SUBJECTS 

58. Tables of Rules for Functions of an Angle 



sin 


opposite side 
- , : side opp. 
hyp. 


= sin X hyp. 


, side opp. 

hyp. = .--^ 

-^^ sin 


cos 


adjacent side 
- , : side adi. 
hyp. 


= cos X hyp. 


side adj. 

: hyp. = - 

-^^ cos 



opposite side . , ^ . , ,. . , 

tan = -^. 7 — ^'. side opp. = tan X side adj.: side adj. = 

side opp. 
tan 

adjacent side . , ,. , . , . , 

cot = r, r-j-: side adi. = cot X side opp.: side opp. = 

opposite side •" ^^ ^^ 

side adj. 
cot 



opp. = opposite 
hyp. = hypothenuse 
adj. = adjacent 





^ 


cj^ 




_> 


^^0^30^ 


* 



59. Calculation of Functions for 30° and 45°.— Refer- 
ring to Fig. 107, in which angle a = 30°, angle h = 60°, 
side A = 10, B = 17.3, and C = 20. 



3-/7.3 
Fig. 107. 

• ono opp. s ide 10 

sm a = sm 30 = — r = 7^ = -^ 

hyp. 20 

o adj. side 17.3 

cos a = COS. 30 = — f = -.^^ = .866 

hyp. 20 

or^o opp. side 10 ^__ 

tana = tan 30° = -~^ — r^ = t^^t^ = -577 

adj. side 17.3 

- ^^o adj. side 17.3 ^ „_ 
cot a = cot 30° = — ^ — ^T- = r^ = 1.73 

opp. side 10 



MEASUREMENT OF RIGHT TRIANGLES 131 



In the same way for Fig. 108, in which a = 45°, 6 = 
45°, A = 20, B = 20, and C = 28.3 



sin a = sin 45° = 



cos a = cos 45° = 



opp. side _ 20 
~li3^7" ~ 2873 
adj. side _ 20 
hyp. ~ 28.3 



= .707 



= .707 



Ar-o <^PP- side 20 ^ ^_ 

tan a = tan 45° = -T^ r- = ^ = 1 . 00 



adj. side 20 



^ ,^o adj. side 20 
cot a = cot 45 = 



opp. side 20 



= 1.00 




5-- 20 
Fig. 108. 



60. Explanation of Use of Tables. — These functions, 
or ratios of sides for a right triangle can be calculated for 
any given angle by constructing the right triangle and 
measuring the length of its sides, but tables are prepared 
which give us these ratios for all angles from 0° to 90° 
in minute steps (see pages 133 to 156). These are the 
tables which we use in making calculations. In using 
these tables the angle in degrees is read in the horizontal 
line at the top of the page for angles up to 44° and the 
minutes in the vertical columns to the left. Angles of 
45° or over are read in the horizontal line at the bottom 
of the page for degrees and the minutes in the verti- 



132 MECHANICS AND ALLIED SUBJECTS 

cal columns to the right. Thus page 134 gives by 
minute steps, the sines and cosines of all angles, from 1° 
to 4° inclusive and the sines and cosines of all angles from 
85° to 88° inclusive. Page 135 gives by minute steps 
the sines and cosines of all angles from 5° to 8° inclusive 
and the sines cosines of all angles from 81° to 84° inclusive. 
Page 145 gives the tangents and cotangents of angles 
from 0° to 3° and the tangents and contangents of angles 
from 86° to 89° and page 146 gives the tangeiits and 
cotangents of angles from 4° to 7° and tangents and 
cotangents of angles from 82° to 85°. Referring to 
page 137. 

The sine of 14° 30' = .25038 
The sine of 13° 20' = .23062 
The sine of 16° 40' = .28680 

From page 143 the cosine of 50° 20' = .63832 (use the 
right-hand column of minutes and degrees at the hottoin 
of the page). 

The cosine of 73° 50' = .27843. The cosine of 0° 15' 
= .99999. The tables of tangents and cotangents, pages 
145 to 146 are read in the same way. 

PROBLEMS 

1. From the tables of sines and cosines find the sines of the 
following angles: 15°, 30°, 60°, 90° (that is 89° 60'), 32° 10', 43° 42', 
2° 4', 44° 50', 40° 55', 45°. 

2. Find the cosines of the same angles given in Prob. 1. 

3. From the table of tangents and contangents find the tangents 
of the following angles: 14° 10', 27° 57', 42° 18', 53° 12', 63', 28°, 
45°, 30°, 17° 27', 36°. 

4. Find the cotangents of the same angles given in Prob. 3. 

5. Using the tables of functions complete the following table, 
finding first the angle from the function given, and then knowing 
the angle, find the other functions. 



MEASUREMENT OF RIGHT TRIANGLES 133 



Prob. No. 


Angle 


Sine 


Cosine 


Tangent 


Cotangent 


1 




.21644 








2 






.48226 






3 




.82248 








4 






.87462 






5 








6.31375 




6 










1 . 12369 



61. Tables of Natural Sines, Cosines, Tangents, and 
Cotangents of Angles from 0° to 90° Varying by Minute 
Angles. 





( 


r 






c 


)° 






0° 




/ 


Sine 


Cosine 


60 


f 

21 


Sine 


Cosine 


39 


41 


Sine 


Cosine 


f 


o 


•OOOOO 




.00611 


•99998 


.01193 


•99993 


iq 


I 


.00029 




59 


22 


.00640 


.99998 


38 


42 


.01222 


•99993 


18 


2 


.00058 




58 


23 


.00669 


.99998 


37 


43 


.01251 


•99992 


17 


3 


.00087 




57 


24 


.00698 


.99998 


36 


44 


.01280 


.99992 


16 


4 


.00116 




56 


25 


.00727 


•99997 


35 


45 


.01309 


.99991 


15 


5 


.00145 




55 


26 


.00756 


.99997 


34 


46 


.01338 


.99991 


14 


6 


.00175 




54 


27 


•00785 


.99997 


33 


47 


.01367 


.99991 


13 


7 


.00204 




53 


28 


.00814 


.99997 


32 


48 


•01396 


.99990 ' 


12 


8 


.00233 




52 


29 


.00844 


•99996 


31 


49 


•01425 


.99990 


11 


9 


.00262 




51 


30 


.00873 


.99996 


30 


50 


.01454 


.99989 


10 


lO 


.00291 




50 


31 


.00902 


.99996 


29 


51 


.01483 


•99989 


Q 


11 


.00320 


•99999 


49 


32 


.00931 


.99996 


28 


52 


•01513 


.99989 


8 


12 


.00349 


.99999 


48 


33 


.00960 


.99995 


27 


53 


.01542 


.99988 


7 


A3 


.00378 


•99999 


47 


34 


.00989 


.99995 


26 


54 


.01571 


.99988 


6 


14 


.00407 


.99999 


46 


35 


.01018 


.99995 


25 


55 


.01600 


.99987 


5 


15 


.00436 


•99999 


45 


36 


.01047 


•99995 


24 


56 


.01629 


•99987 


4 


i6 


.00465 


.99999 


44 


37 


.01076 


.99994 


23 


57 


.01658 


.99986 


3 


17 


.00495 


.99999 


43 


38 


.01105 


.99994 


22 


58 


.01687 


.99986 


2 


i8 


.00524 


•99999 


42 


39 


.01134 


.99994 


21 


59 


.01716 


.99985 


I 


19 


•00553 


•99998 


41 


40 


.01164 


•99993 


20 


60 


•01745 


.99985 





20 


.00582 


.99998 


40 








/ 


/ 








/ 


Cosine 


Sine 


f 


/ 


Cosine 


Sine 


Cosine 


Sine 


/ 




8 


9° 






8< 


r 






8{ 


)^ 





134 



MECHANICS AND ALLIED SUBJECTS 



O 

I 
2 

3 
4 
5 
6 
7 
8 

9 

lo 

II 

12 

13 
14 
15 
i6 
17 
i8 
19 

20 
21 
22 

23 

24 

25 
26 
27 
28 
29 
30 

31 
32 

33 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



1° 




2° 




«. 


;0 


40 




Sine ( 


"OSINE 


Sine ( 


"osine 


Sine 


Cosine 


Sine 


Cosine 


/ 


•01745 


99985 


•03490 


99939 


•05234 


•99863 


.06976 


.99756 


60 


.01774 


99984 


.03519 


99938 


.05263 


.99861 


.07005 


•99754 


S9 


.01803 


99984 


.03548 


99937 


•05292 


.99860 


.07034 


•99752 


58 


.01832 


99983 


.03577 


99936 


•05321 


.99858 


.07063 


•99750 


57 


.01862 


99983 


.03606 


99935 


•05350 


.99857 


.07092 


.99748 


56 


.01891 


99982 


.03635 


99934 


•05379 


.99855 


.07121 


.99746 


55 


.01920 


99982 


.03664 


99933 


.05408 


.99854 


.07150 


.99744 


54 


.01949 


99981 


.03693 


99932 


•05437 


.99852 


.07179 


.99742 


53 


.01978 


99980 


•03723 


99931 


.05466 


.99851 


.07208 


.99740 


52 


.02007 


99980 


.03752 


99930 


•05495 


.99849 


•07237 


•99738 


51 


.02036 


99979 


.03781 


99929 


.05524 


.99847 


.07266 


•99736 


50 


.02065 


99979 


.03810 


99927 


•05553 


.99846 


.07295 


.99734 


49 


.02094 


99978 


•03839 


99926 


•05582 


.99844 


•07324 


•99731 


48 


.02123 


99977 


.03868 


99925 


.05611 


.99842 


•07353 


.99729 


47 


.02152 


99977 


.03897 


99924 


.05640 


.99841 


.07382 


.99727 


46 


.02181 


99976 


.03926 


99923 


.05669 


.99839 


.07411 


.99725 


45 


.02211 


99976 


.03955 


99922 


.05698 


.99838 


.07440 


.99723 


44 


.02240 


99975 


.03984 


99921 


.05727 


.99836 


.07469 


.99721 


43 


.02269 


99974 


.04013 


99919 


.05756 


.99834 


.07498 


.99719 


42 


.02298 


99974 


.04042 


99918 


•05785 


•99833 


•07527 


.99716 


41 


.02327 


99973 


.04071 


99917 


•05814 


.99831 


.07556 


.99714 


40 


.02356 


99972 


.04100 


99916 


.05844 


.99829 


.07585 


.99712 


39 


.02385 


99972 


.04129 


99915 


.05873 


.99827 


.07614 


.99710 


38 


.02414 


99971 


.04159 


99913 


.05902 


.99826 


.07643 


.99708 


37 


.02443 


99970 


.04188 


99912 


.05931 


.99824 


.07672 


•99705 


36 


.02472 


99969 


.04217 


9991 1 


.05960 


.99822 


.07701 


•99703 


35 


.02501 


99969 


.04246 


99910 


.05989 


.99821 


.07730 


.99701 


34 


.02530 


99968 


.04275 


99909 


.06018 


.99819 


•07759 


.99699 


33 


.02560 


99967 


.04304 


99907 


.06047 


.99817 


.07788 


.99696 


32 


.02589 


99966 


.04333 


99906 


.06076 


.99815 


.07817 


•99694 


31 


.02618 


99966 


.04362 


99905 


.06105 


.99813 


.07846 


.99692 


30 


.02647 


99965 


.04391 


99904 


.06134 


.99812 


•07875 


.99689 


29 


.02676 


99964 


.04420 


99902 


.06163 


.99810 


.07904 


.99687 


23 


.02705 


99963 


.04449 


99901 


.06192 


.99808 


•07933 


.99685 


27 


.02734 


99963 


.04478 


99900 


.06221 


.99806 


.07962 


.99683 


26 


.02763 


99962 


.04507 


99898 


.06250 


.99804 


•07991 


.99680 


25 


.02792 


99961 


.04536 


99897 


.66279 


.99803 


.08020 


.99678 


24 


.02821 


99960 


.04565 


99896 


.06308 


.99801 


.08049 


.99676 


23 


.02850 


99959 


.04594 


99894 


•06337 


•99799 


.08078 


•99673 


22 


.02879 


99959 


.04623 


99893 


.06366 


.99797 


.08107 


.99671 


21 


.02908 


99958 


.04653 


99892 


•06395 


.99795 


.08136 


.99668 


20 


.02938 


99957 


.04682 


99890 


.06424 


•99793 


.08165 


.99666 


19 


.02967 


99956 


.04711 


99889 


.06453 


.99792 


.08194 


.99664 


18 


.02996 


99955 


.04740 


99888 


.06482 


.99790 


.08223 


.99661 


17 


.03025 


99954 


.04769 


99886 


.06511 


.99788 


.08252 


.99659 


16 


.03054 


99953 


.04798 


99885 


.06540 


.99786 


.08281 


•99657 


15 


.03083 


99952 


.04827 


99883 


.06569 


.99784 


.08310 


.99654 


14 


.03112 


99952 


.048^6 


99882 


.06598 


.99782 


.08339 


.99652 


13 


.03141 


99951 


.04885 


99881 


.06627 


.99780 


.08368 


.99649 


12 


.03170 


99950 


.04914 


99879 


.06656 


.99778 


.08397 


.99647 


II 


.03199 


99949 


•04943 


99878 


.06685 


.99776 


.08426 


.99644 


10 


.03228 


99948 


.04972 


99876 


.06714 


•99774 


.08455 


.99642 


9 


•03257 


99947 


.05001 


99875 


.06743 


•99772 


.08484 


.99639 


8 


.03286 


99946 


.05030 


99873 


.06773 


.99770 


•08513 


•99637 


7 


.03316 


99945 


•05059 


99872 


.06802 


.99768 


.08542 


•99635 


6 


•03345 


99944 


.05088 


99870 


.06831 


.99766 


•08571 


•99632 


5 


.03374 


99943 


.05117 


99869 


.06860 


.99764 


.08600 


.99630 


4 


.03403 


99942 


.05146 


99867 


.06889 


.99762 


.08629 


.99627 


3 


.03432 


99941 


•05175 


99866 


.06918 


.99760 


.08658 


.99625 


2 


.03461 


99940 


•0520s 


.99864 


.06947 


•99758 


.08687 


.99622 


I 


•03490 


99939 


•05234 


.99863 


.06976 


•99756 


.08716 


.99619 





Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


f 


88° 




87° 




8( 


3° 


& 


5° 





MEASUREMENT OF RIGHT TRIANGLES 



135 





t 


5° 


( 


r 


70 1 


8° 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


Cosine 


o 


.08716 


.99619 


.10453 


.99452 


.12187 


•99255 


.13917 


99027 


I 


.08745 


.99617 


.10482 


.99449 


.12216 


•99251 


.13946 


99023 


2 


.08774 


.99614 


.10511 


.99446 


.12245 


.99248 


.13975 


99019 


3 


.08803 


.99612 


.10540 


.99443 


.12274 


.99244 


.14004 


99015 


4 


.08831 


.99609 


.10569 


.99440 


.12302 


.99240 


•14033 


99011 


5 


.08860 


.99607 


.10597 


.99437 


•12331 


•99237 


.14061 


99006 


6 


.08889 


.99604 


.10626 


.99434 


.12360 


•99233 


.14090 


99002 


7 


.08918 


.99602 


.10655 


•99431 


.12389 


.99230 


.14119 


98998 


8 


.08947 


.99599 


.10684 


.99428 


.12418 


.99226 


.14148 


98994 


9 


.08976 


.99596 


.10713 


.99424 


.12447 


.99222 


.14177 


.98990 


lO 


.09005 


•99594 


.10742 


.99421 


.12476 


.99219 


.14205 


.98986 


II 


.09034 


•99591 


.10771 


.99418 


•12504 


.99215 


.14234 


98982 


12 


.09063 


.99588 


.10800 


.99415 


•12533 


.99211 


.14263 


.98978 


13 


.09092 


.99586 


.10829 


.99412 


.12562 


.99208 


.14292 


98973 


14 


.09121 


.99583 


.10858 


.99409 


.12591 


.99204 


.14320 


98969 


15 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200 


•14349 


.98965 


i6 


.09179 


•99578 


.10916 


.99402 


.12649 


.99197 


•14378 


98961 


17 


.09208 


•99575 


,10945 


•99399 


.12678 


.99193 


•14407 


.98957 


i8 


.09237 


•99572 


.10973 


.99396 


.12706 


.99189 


•14436 


.98953 


19 


.09266 


.99570 


.11002 


•99393 


•12735 


.99186 


.14464 


.98948 


20 


.09295 


.99567 


.11031 


.99390 


.12764 


.99182 


.14493 


.98944 


21 


.09324 


.99564 


.11060 


.99386 


.12793 


.99178 


.14522 


.98940 


22 


•09353 


.99562 


.11089 


.99383 


.12822 


•99175 


.14551 


.98936 


23 


.09382 


.99559 


.11118 


.99380 


.i28=;i 


.99171 


.14580 


.98931 


24 


.09411 


.99556 


.11147 


.99377 


.12880 


•99167 


.14608 


.98927 


25 


.09440 


.99553 


.11176 


•99374 


.12908 


•99163 


.14637 


.98923 


26 


.09469 


.99551 


.11205 


•99370 


.12937 


.99160 


.14666 


.98919 


^Z 


.09498 


.99548 


.11234 


•99367 


.12966 


•99156 


.14695 


.98914 


28 


.09527 


•99545 


.11263 


.99364 


.12995 


.99152 


.14723 


.98910 


29 


.09556 


.99542 


.11291 


.99360 


• 13024 


.99148 


.14752 


.98906 


30 


•09585 


.99540 


.11320 


•99357 


.13053 


•99144 


.14781 


.98902 


31 


.09614 


.99537 


.11349 


•99354 


.13081 


.99141 


.14810 


98897 


32 


.09642 


•99534 


.11378 


•99351 


.13110 


•99137 


.14838 


.98893 


33 


.09671 


•99531 


.11407 


•99347 


.13139 


.99133 


.14867 


.98889 


34 


.09700 


.99528 


.11436 


•99344 


.13168 


.99129 


.14896 


98884 


2| 


.09729 


•99526 


.11465 


•99341 


.13197 


•99125 


.14925 


98880 


36 


.09758 


.99523 


.11494 


•99337 


.13226 


.99122 


.14954 


98876 


^2 


.09787 


•99520 


.11523 


•99334 


.13254 


.99118 


.14982 


98871 


38 


.09816 


•99517 


.11552 


•99331 


.13283 


.99114 


.15011 


98867 


39 


.09845 


.99514 


.11580 


•99327 


•13312 


.99110 


.15040 


98863 


40 


.09874 


•995 II 


.11609 


•99324 


.13341 


.99106 


.15069 


98858 


41 


.09903 


.99508 


.11638 


.99320 


.13370 


.99102 


.15097 


98854 


42 


.09932 


•99506 


.11667 


•99317 


•13399 


.99098 


.15126 


98849 


43 


.09961 


•99503 


.11696 


•99314 


•13427 


•99094 


.15155 


9884s 


44 


.09990 


.99500 


.11725 


.99310 


•13456 


.99091 


.15184 


98841 


45 


.10019 


.99497 


.11754 


•99307 


•13485 


.99087 


.15212 


98836 


46 


.10048 


.99494 


.11783 


.99303 


•13514 


.99083 


.15241 


9?532 


47 


.10077 


.99491 


.11812 


.99300 


•13543 


•99079 


.15270 


98827 


48 


.10106 


.99488 


.11840 


•99297 


•13572 


•99075 


.15299 


98823 


49 


.10135 


.99485 


.11869 


•99293 


.13600 


•99071 


.15327 


98818 


50 


.10164 


.99482 


.11898 


.99290 


.13629 


.99067 


.15356 


98814 


51 


.10192 


.99479 


.11927 


.99286 


•13658 


.99063 


.15385 


98809 


52 


.10221 


.99476 


.11956 


.99283 


.13687 


.99059 


.15414 


98805 


53 


.10250 


.99473 


.11985 


.99279 


.13716 


•99055 


.15442 


98800 


54 


.10279 


.99470 


.12014 


.99276 


.13744 


.99051 


.15471 


98796 


55 


.10308 


.99467 


.12043 


.99272 


.13773 


.99047 


.15500 


98791 


56 


.10337 


.99464 


.12071 


.99269 


.13802 


.99043 


.15529 


98787 


57 


.10366 


.99461 


.12100 


•99265 


.13831 


•99039 


.15557 


98782 


58 


•10395 


.99458 


.12129 


.99262 


.13860 


•99035 


.15586 


98778 


59 


.10424 


.99455 


.12158 


.99258 


.13889 


.99031 


.15615 . 


98773 


60 


.10453 


.99452 


.12187 


.99255 


.13917 


.99027 


.15643 


98769 


f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 1 


Sine 




8^ 


t° 


8: 


5° 


& 


>o 


81° 





60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

3g 
38 

37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 

IS 
14 
13 
12 
II 
10 

9 
8 

7 
6 

5 
4 
3 
2 

I 
o 



136 



MECHANICS AND ALLIED SUBJECTS 





9° 


K 


D° 


1 





120 




t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


r 


o 


.15643 


.98769 


•17365 


.98481 


.19081 


.98163 


.20791 


.97815 


60 


I 


.15672 


.98764 


•17393 


.98476 


.19109 


.98157 


.20820 


.97809 


59 


2 


.15701 


.98760 


.17422 


.98471 


.19138 


.98152 


.20848 


.97803 


58 


3 


.15730 


.98755 


.17451 


.98466 


.19167 


.98146 


.20877 


.97797 


57 


4 


.15758 


.98751 


.17479 


.98461 


•19195 


.98140 


.20905 


•97791 


56 


5 


.15787 


.98746 


.17508 


•98455 


.19224 


.98135 


.20933 


•97784 


55 


6 


.15816 


.98741 


.17537 


•98450 


.19252 


.98129 


.20962 


•97778 


54 


7 


.15845 


•98737 


.17565 


.98445 


.19281 


.98124 


.20990 


.97772 


53 


8 


.15873 


.98732 


.17594 


.98440 


.19309 


.98118 


.21019 


.97766 


52 


9 


.15902 


.98728 


.17623 


•98435 


.19338 


.98112 


.21047 


.97760 


51 


lO 


•15931 


.98723 


.17651 


•98430 


.19366 


.98107 


.21076 


.97754 


50 


II 


•15959 


.98718 


.17680 


.98425 


.19395 


.98101 


.21104 


•97748 


49 


12 


.15988 


.98714 


.17708 


.98420 


.19423 


.98096 


.21132 


•97742 


48 


13 


.16017 


.98709 


•17737 


.98414 


.19452 


.98090 


.21161 


•97735 


47 


14 


.16046 


.98704 


.17766 


.98409 


.19481 


.98084 


.21189 


•97729 


46 


15 


.16074 


.98700 


•17794 


.98404 


.19509 


.98079 


.21218 


.97723 


45 


i6 


.16103 


.98695 


.17823 


.98399 


•19538 


.98073 


.21246 


.97717 


44 


17 


.16132 


.98690 


.17852 


.98394 


.19566 


.98067 


.21275 


.97711 


43 


i8 


.16160 


.98689 


.17880 


.98389 


•19595 


.98061 


.21303 


.97705 


42 


19 


.16189 


.98681 


.17909 


•98383 


•19623 


.98056 


•21331 


.97698 


41 


20 


.16218 


.98676 


•17937 


•98378 


.19652 


.98050 


.21360 


■ .97692 


40 


21 


.16246 


.98671 


.17966 


•98373 


.19680 


.98044 


.21388 


.97686 


39 


22 


.16275 


.98667 


•17995 


.98368 


.19709 


•98039 


.21417 


.97680 


38 


23 


.16304 


.98662 


.18023 


.98362 


•19737 


•98033 


.21445 


.97673 


37 


24 


.16333 


.98657 


.18052 


•98357 


.19766 


.98027 


.21474 


.97667 


36 


25 


.16361 


.98652 


.18081 


•98352 


.19794 


.98021 


.21502 


.97661 


35 


26 


.16390 


.98648 


.18109 


.98347 


.19823 


.98016 


.21530 


.97655 


34 


27 


.16419 


.98643 


.18138 


.98341 


•19851 


.98010 


.21559 


.97648 


ZZ 


28 


.16447 


.98638 


.18166 


•98336 


.19S80 


.98004 


.21587 


.97642 


32 


29 


.16476 


.98633 


.18195 


•98331 


.19908 


.97987 


.21616 


.97636 


31 


30 


.16505 


.98629 


.18224 


•98325 


.19937 


•97992 


.21644 


.97630 


30 


31 


.16533 


.98624 


.18252 


.98320 


.19965 


.97987 


.21672 


.97623 


29 


32 


.16562 


.98619 


.18281 


.98315 


.19994 


.97981 


.21701 


.97617 


28 


33 


.16591 


.98614 


.18309 


.98310 


.20022 


•97975 


.21729 


.97611 


27 


34 


.16620 


.98609 


.18338 


.98304 


.20031 


.97969 


.21758 


.97604 


26 


35 


.16648 


.98604 


•18367 


.98299 


.20079 


.97963 


.21786 


.97598 


25 


36 


.16677 


.98600 


•18395 


.98294 


.20108 


•97958 


.21814 


.97592 


24 


37 


.16706 


•98595 


.18424 


.98288 


.20136 


•97952 


.21843 


.97585 


23 


38 


.16734 


.98590 


.18452 


•98283 


.20165 


.97946 


.21871 


.97579 


22 


39 


.16763 


.98585 


.18481 


.98277 


.20193 


•97940 


.21899 


•97573 


21 


40 


.16792 


.98580 


.18509 


.98272 


.20222 


•97934 


.21928 


•97566 


20 


41 


.16820 


.98575 


.18538 


.98267 


.20250 


.97928 


.21956 


•97560 


19 


42 


.16849 


.98570 


.18567 


.98261 


.20279 


.97922 


•21985 


•97553 


18 


43 


.16878 


.98565 


.18595 


.98256 


.20307 


.97916 


.22013 


•97547 


17 


44 


.16906 


.98561 


.18624 


.98250 I 


.20336 


.97910 


.22041 


•97541 


16 


45 


.16935 


•98556 


.18652 


.98245 


.20364 


•97905 


.22070 


•97534 


15 


46 


.16964 


.98551 


.18681 


.98240 


.20393 


•97899 


.22098 


•97528 


14 


47 


.16992 


.98546 


.18710 


.98234 


.20421 


•97893 


.22126 


.97521 


13 


48 


.17021 


.98541 


•18738 


.98229 


.20450 


.97887 


.22155 


.97515 


12 


49 


.17050 


.98536 


.18767 


.98223 


.20478 


.97881 


.22183 


.97508 


II 


50 


.17078 


•98531 


•18795 


.98218 


.20507 


.97875 


.22212 


.97502 


10 


51 


.17107 


.98526 


.18824 


.98212 


•20535 


.97869 


.22240 


.97496 


9 


52 


.17136 


.98521 


.18852 


.98207 


•20563 


•97863 


.22268 


.97489 


8 


53 


.17164 


.98516 


.18881 


.98201 


.20592 


•97857 


.22297 


.97483 


7 


54 


.17193 


.98511 


.18910 


.98196 


.20620 


•97851 


.22325 


.97476 


6 


55 


.17222 


.98506 


.18938 


.98190 


.20649 


.97845 


.22353 


.97470 


5 


56 


.17250 


•98501 


.18967 


.98185 


.20677 


•97839 


.22382 


.97463 


4 


57 


.17279 


.98496 


.18995 


.98179 


.20706 


.97833 


.22410 


.97457 


3 


58 


.17308 


.98491 


.19024 


.98174 


.20734 


.97827 


.22438 


.97450 


2 


59 


.17336 


.98486 


.19052 


.98168 


.20763 


.97821 


.22467 


.97444 


I 


60 


.17365 


.98481 


.19081 


.98163 


.20791 


.97815 


.22495 


^7437 


o 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sj[NE 


/ 




8C 


)° 


7S 


|o 


78 


1° 


77 


TO 





MEASUREMENT OF RIGHT TRIANGLES 



137 





13° 


140 


15° 


18° 


1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


/ 


o 


.22495 


•97437 


.24192 


•97030 


.25882 


.96593 


.27564 


.96126 


6c 


I 


.22523 


•97430 


.24220 


•97023 


.25910 


•96585 


.27592 


.96118 


59 


2 


.22552 


•97424 


.24249 


•97015 


.25938 


.96578 


.27620 


.96110 


58 


3 


.22580 


•97417 


.24277 


.97008 


.25966 


.96570 


.27648 


.96102 


57 


4 


.22608 


.97411 


.24305 


.97001 


.25994 


.96562 


.27676 


.96094 


S6 


5 


.22637 


.97404 


.24333 


.96994 


.26022 


•96555 


.27704 


.96086 


55 


6 


.22665 


.97398 


.24362 


.96987 


.26050 


•96547 


.27731 


.96078 


54 


7 


.22693 


•97391 


.24390 


.96980 


.26079 


•96540 


•27759 


.96070 


53 


8 


.22722 


•97384 


.24418 


.96973 


.26107 


•96532 


•27787 


.96062 


52 


9 


.22750 


•97378 


.24446 


.96966 


•26135 


•96524 


•27815 


•96054 


51 


lO 


.22778 


•97371 


.24474 


•96959 


.26163 


•96517 


•27843 


.96046 


50 


II 


.22807 


•97365 


.24503 


•96952 


.26191 


.96509 


.27871 


.96037 


49 


12 


.22835 


•97358 


.24531 


.96945 


.26219 


.96502 


.27899 


.96029 


48 


13 


.22863 


.97351 


.24559 


•96937 


.26247 


.96494 


.27927 


.96021 


47 


14 


.22892 


.97345 


.24587 


.96930 


.26275 


.96486 


•27955 


.96013 


46 


IS 


.22920 


.97338 


.24615 


.96923 


•26303 


.96479 


.27983 


.96005 


45 


i6 


.22948 


.97331 


.24644 


.96916 


•26331 


.96471 


.28011 


.95997 


44 


17 


.22977 


.97325 


.24672 


.96909 


.26359 


.96463 


.28039 


.95989 


43 


i8 


.23005 


.97318 


.24700 


.96902 


.26387 


.96456 


.28067 


.95981 


42 


19 


.23033 


.97311 


.24728 


.96894 


.26415 


.96448 


.28095 


•95972 


41 


20 


.23062 


•97304 


.24756 


.96887 


.26443 


.96440 


.28123 


.95964 


40 


21 


.23090 


.97298 


.24784 


.96880 


.26471 


•96433 


.28150 


.95956 


39 


22 


.23118 


.97291 


.24813 


.96873 


.26500 


.96425 


.28178 


.95948 


38 


23 


.23146 


.97284 


.24841 


.96866 


.26528 


.96417 


.28206 


.95940 


37 


24 


.23175 


•97278 


.24869 


.96858 


.26556 


.96410 


.28234 


.95931 


36 


25 


.23203 


.97271 


.24897 


.96851 


.26=584 


.96402 


.28262 


•95923 


35 


26 


.23231 


•97264 


•24925 


.96844 


.26612 


.96394 


.28290 


.95915 


34 


27 


.23260 


•97257 


.24954 


.96837 


.26640 


.96386 


.28318 


.95007 


33 


28 


.23288 


•97251 


.24982 


.96829 


.26668 


.96379 


.28346 


.95898 


32 


29 


.23316 


•97244 


.25010 


.96822 


.26696 


•96371 


.28374 


.95890 


31 


30 


•23345 


•97237 


•25038 


.96815 


.26724 


•96363 


.28402 


.95882 


30 


31 


.23373 


•97230 


.25066 


.96807 


.26752 


•96355 


.28429 


•95874 


29 


32 


.23401 


.97223 


.25094 


.96800 


.26780 


•96347 


•28457 


•95865 


28 


33 


.23429 


.97217 


.25122 


.96793 


.26808 


•96340 


.28485 


.95857 


27 


34 


.23458 


.97210 


.25151 


.96786 


.26836 


.96332 


•28513 


.95849 


26 


35 


.23486 


.97203 


.25179 


•96778 


.26864 


•96324 


.28541 


.95841 


25 


36 


.23514 


.97196 


.25207 


.96771 


.26892 


.96316 


.28569 


•95832 


24 


37 


•23542 


.97189 


•25235 


.96764 


.26920 


.96308 


.28597 


.95824 


23 


38 


.23571 


.97182 


.25263 


.96756 


.26948 


.96301 


.28625 


.95816 


22 


39 


.23599 


•97176 


.25291 


.96749 


.26976 


•96293 


.28652 


.95807 


21 


40 


.23627 


.97169 


.25320 


.96742 


.27004 


.96285 


.28680 


.95799 


20 


41 


.23656 


.97162 


•25348 


•96734 


.27032 


.96277 


.28708 


•95791 


19 


42 


.23684 


•97155 


•25376 


•96727 


.27060 


.96269 


.28736 


.95782 


i3 


43 


.23712 


.97148 


.25404 


.96719 


.27088 


.96261 


.28764 


.95774 


17 


44 


.23740 


.97141 


•25432 


.96712 


.27116 


•96253 


.28792 


.95766 


16 


45 


.23769 


•97134 


.25460 


.96705 


.27144 


.96246 


.28820 


.95757 


15 


46 


.23797 


.97127 


.25488 


.96697 


.27172 


.96238 


.28847 


.95749 


14 


47 


.23825 


.97120 


•25516 


.96690 


.27200 


.96230 


•28875 


.95740 


13 


48 


.23853 


•97113 


•25545 


.96682 


.27228 


.96222 


.28903 


•95732 


12 


49 


.23882 


.97106 


.25573 


.96675 


.27256 


.96214 


•28931 


•95724 


II 


50 


.23910 


.97100 


.25601 


.96667 


.27284 


.96206 


.28959 


•95715 


10 


51 


.23938 


.97093 


.25629 


.96660 


.27312 


.96198 


.28987 


•95707 


9 


52 


.23966 


.97086 


•25657 


.96653 


.27340 


.96190 


.29015 


.95698 


8 


53 


.23995 


•97079 


.25685 


.96645 


.27368 


.96182 


.29042 


.95690 


7 


54 


.24023 


•97072 


.25713 


.96638 


.27396 


.96174 


.29070 


•95681 


6 


55 


.24051 


.97065 


.25741 


.96630 


.27424 


.96166 


.29098 


•95673 


5 


56 


.24079 


.97058 


.25769 


.96623 


.27452 


.96158 


.29126 


.95664 


4 


57 


.24108 


•97051 


.25798 


.96615 


.27480 


.96150 


.29154 


•95656 


3 


58 


.24136 


•97044 


.25826 


.96608 


.27508 


.96142 


.29182 


•95647 


2 


59 


.24164 


•97037 


.25854 


.96600 


.27536 


.96134 


.29209 


.95639 


I 


60 


.24192 


•97030 


.25882 


•96593 


.27564 


.96126 


.29237 


•95630 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


f 




7€ 


>o 


7^ 


;° 


74 


P 


75 


1° 





138 



MECHANICS AND ALLIED SUBJECTS 





17° 


1 


8° 


19° 


20° 





Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.29237 


.95630 


.30902 


.95106 


.32557 


•94552 


.34202 


.93969 


I 


.29265 


.95622 


.30929 


.95097 


.32584 


.94542 


.34229 


•93959 


2 


.29293 


.95613 


•30057 


.95088 


.32612 


.94533 


•34257 


•93949 


3 


.29321 


•95605 


.30985 


.95079 


.32639 


.94523 


.34284 


•93939 


4 


.29348 


•95596 


.31012 


.95070 


.32667 


•94514 


•34311 


•93929 


5 


.29376 


•95588 


.31040 


.95061 


.32694 


.94504 


•34339 


•93919 


6 


.29404 


•95579 


.31068 


.95052 


.32722 


•94495 


•34366 


.93909 


7 


.29432 


•95571 


•31095 


•95043 


.32749 


•94485 


•34393 


.93899 


8 


.29460 


.95562 


•31123 


•95033 


.32777 


•94476 


.34421 


•93889 


9 


.29487 


•95554 


•31151 


•95024 


.32804 


.94466 


•34448 


•93879 


lO 


.29515 


•95545 


.31178 


•95015 


.32832 


•94457 


.34475 


.93869 


II 


.29543 


•95536 


.31206 


.95006 


•32859 


•94447 


•34503 


•93859 


12 


•29571 


•95528 


■3^233 


.94997 


.32887 


•94438 


•34530 


.93849 


13 


.29599 


•95519 


.31261 


.94988 


.32914 


.94428 


.34557 


•93839 


14 


.29626 


•95511 


.31289 


•94979 


.32942 


.94418 


.34584 


.93829 


15 


.29654 


•95502 


•31316 


•94970 


.32969 


•94409 


.34612 


.93819 


i6 


.29682 


•95493 


•31344 


.94961 


.32997 


•94399 


•34639 


•93809 


17 


.29710 


•95485 


•31372 


•94952 


.33024 


•94390 


.34666 


.93799 


i8 


.29737 


•95476 


•31399 


•94943 


.33051 


•94580 


.34694 


.93789 


19 


.29765 


•95467 


•31427 


•94933 


.33079 


•94370 


.34721 


.93779 


20 


.29793 


•95459 


•31454 


.94924 


.33106 


•94361 


.34748 


•93769 


21 


.29821 


.95450 


•31482 


•94915 


.33134 


•94351 


.34775 


•93759 


22 


.29849 


•95441 


•31510 


.94906 


.33161 


.94342 


.34803 


•93748 


23 


.29876 


•95433 


•31537 


•94897 


•33189 


•94332 


.34830 


•93738 


24 


.29904 


•95424 


•31565 


.94888 


.33216 


.94322 


.34857 


•93728 


25 


.29932 


•95415 


•31593 


.94878 


.53244 


■94313 


.34884 


•93718 


26 


.29960 


•95407 


.31620 


.94869 


.33271 


•94303 


.34912 


•93708 


27 


.29987 


•95398 


•31648 


.94860 


.33298 


•94293 


.34939 


•93698 


28 


.30015 


•95389 


•31675 


•94851 


.33326 


.94284 


.34966 


.93688 


29 


•30043 


•95380 


•31703 


.94842 


'33353 


.94274 


•34993 


•93677 


30 


•30071 


•95372 


•31730 


•94832 


.33381 


.94264 


•35021 


•93667 


31 


.30098 


•95363 


•31758 


.94823 


.33408 


.94254 


•35048 


•93657 


32 


.30126 


•95354 


•31786 


.94814 


.33436 


•94245 


•35075 


•93647 


33 


•30154 


•95345 


.31813 


•94805 


.33463 


•94235 


•35102 


•93637 


34 


.30182 


•95337 


•31841 


•94795 


.33490 


•94225 


•35130 


.93626 


35 


.30209 


•95328 


.31868 


.94786 


.33518 


.94215 


•35157 


•93616 


36 


•30237 


•95319 


•31896 


•94777 


.33545 


.94206 


•35184 


.93606 


37 


•30265 


•95310 


•31923 


.94768 


•33573 


.94196 


•35211 


•93596 


38 


.30292 


•95301 


•31951 


•94758 


.33600 


.94186 


•35239 


•93585 


39 


.30320 


•95293 


•31979 


.94749 


•33627 


•94176 


.35266 


•93575 


40 


•30348 


•95284 


.32006 


.94740 


•33655 


•94167 


•35293 


•93565 


41 


•30376 


•95275 


•32034 


•94730 


•33682 


•94157 


•35320 


•93555 


42 


•30403 


•95266 


.32061 


.94721 


.33710 


.94147 


.35347 


•93544 


43 


•30431 


•95257 


.32089 


.94712 


.33737 


.94137 


.35375 


•93534 


44 


•30459 


.95248 


.32116 


.94702 


.33764 


.94127 


.35402 


•93524 


45 


.30486 


•95240 


•32144 


•94693 


.33792 


.94118 


•35429 


•93514 


46 


•30514 


•95231 


.32171 


.94684 


.33819 


.94108 


•35456 


•93503 


47 


•30542 


.95222 


.32199 


•94674 


.33846 


.94098 


•35484 


•93493 


48 


.30570 


•95213 


•32227 


•94665 


.33874 


.94088 


•35511 


•93483 


49 


•30597 


.95204 


•32254 


.94656 


.33901 


.94078 


•35538 


•93472 


50 


•30625 


•95195 


.32282 


.94646 


.33929 


.94068 


•35565 


•93462 


51 


•30653 


.95186 


•32309 


.94637 


•33956 


•94058 


•35592 


•93452 


52 


.30680 


.95177 


•32337 


.94627 


•33983 


•94049 


•35619 


•93441 


53 


•30708 


•95168 


•32364 


.94618 


•34011 


•94039 


•35647 


•93431 


54 


.30736 


•95159 


•32392 


.94609 


•34038 


.94029 


•35674 


•93420 


55 


•30763 


•95150 


•32419 


.94599 


•34065 


.94019 


•35701 


.93410 


56 


•30791 


•95142 


.32447 


.94590 


•34093 


.94009 


•35728 


.93400 


57 


•30819 


•95133 


•32474 


.94580 


.34120 


•93999 


•35755 


•93389 


58 


.30846 


.95124 


•32502 


^4571 


•34147 


•93989 


.35782 


•93379 


59 


.30874 


•95115 


•32529 


^4561 


•34175 


•93979 


.35810 


.93368 


60 


.30902 


.95106 


.32557 


.94552 


.34202 


.93969 


.35837 


•93358 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 




7:: 


)0 


7] 


L° 


7( 


r 


61 


r 



60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 

41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 

28 
27 
26 

25 
24 

23 
22 
21 
20 

19 

18 

17 
16 
15 
14 
13 
12 
II 
10 

9 
8 

7 
6 

5 
4 
3 

2 
I 



MEASUREMENT OF RIGHT TRIANGLES 



139 



2 


V 


22° 


23° 


24° 




Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


/ 


.35837 


•93358 


.37461 


.92718 


.39073 


.92050 


.40674 


.91355 


60 


.35864 


•93348 


.37488 


.92707 


.39100 


.92039 


.40700 


•91343 


59 


.35891 


•93337 


.37515 


.92697 


.39127 


.92028 


.40727 


.91331 


58 


.35918 


•93327 


•37542 


.92686 


.39153 


.92016 


•40753 


•91319 


57 


.35945 


•93316 


.37569 


.92675 


.39180 


.92005 


.40780 


•91307 


56 


.35973 


•93306 


•37595 


.92664 


.39207 


.91994 


.40806 


.91295 


55 


.36000 


•93295 


.37622 


•92653 


•39234 


.91982 


.40833 


.91283 


54 


.36027 


.93285 


.37649 


.92642 


.39260 


.91971 


.40860 


.91272 


53 


.36054 


•93274 


.37676 


.92631 


•39287 


•91959 


.40886 


.91260 


52 


.36081 


•93264 


•37703 


.92620 


.39314 


.91948 


.40913 


.91248 


51 


.36108 


•93253 


•37730 


.92609 


•39341 


.91936 


.40939 


.91236 


50 


.36135 


•93243 


•37757 


.92598 


.39367 


.91925 


.40966 


.91224 


49 


.36162 


•93232 


.37784 


.92587 


•39394 


.91914 


.40992 


.91212 


48 


.36190 


.93222 


.37811 


•92576 


.39421 


.91902 


.41019 


.91200 


47 


.36217 


.93211 


.37838 


•92565 


.39448 


.91891 


.41045 


.91188 


46 


.36244 


.93201 


•37865 


•92554 


.39474 


.91879 


.41072 


.91176 


45 


.36271 


.93190 


•37892 


•92543 


•39501 


.91868 


.41098 


.91164 


44 


.36298 


.93180 


•37919 


•92532 


•39528 


•91856 


.41125 


.91152 


43 


.36325 


.93169 


•37946 


.92521 


•39555 


.91845 


.41151 


.91140 


42 


.36352 


•93159 


•37973 


.92510 


•39581 


.91833 


.41178 


.91128 


41 


.36379 


.93148 


•37999 


.92499 


.39608 


.91822 


.41204 


.91116 


40 


.36406 


•93137 


.38026 


.92488 


.39635 


.91810 


.41231 


.91104 


39 


•36434 


•93127 


•38053 


.92477 


.39661 


.91799 


.41257 


.91092 


38 


•36461 


.93116 


.38080 


.92466 


.39688 


.91787 


.41284 


.91080 


37 


.36488 


.93106 


.38107 


•92455 


.39715 


.91775 


.41310 


.91068 


36 


•36515 


•93095 


•38134 


•92444 


.39741 


.91764 


•41337 


.91056 


35 


•36542 


.93084 


.38161 


.92432 


.39768 


.91752 


.41363 


.91044 


34 


•36569 


.93074 


.38188 


.92421 


.39795 


.91741 


.41390 


.91032 


33 


.36596 


•93063 


•38215 


.92410 


.39822 


.91729 


.41416 


.91020 


32 


.36623 


•93052 


.38241 


.92399 


.39848 


.91718 


.41443 


.91008 


31 


.36650 


•93042 


.38268 


.92388 


.39875 


.91706 


.41469 


.90996 


30 


.36677 


•93031 


.38295 


•92377 


.39902 


.91694 


.41496 


.90984 


29 


•36704 


•93020 


.38322 


.92366 


.39928 


.91683 


.41522 


.90972 


28 


.36731 


.93010 


•38349 


•92355 


•39955 


.91671 


.41549 


.90960 


27 


•36758 


.92999 


•38376 


•92343 


.39982 


.91660 


•41575 


.90948 


26 


•3678s 


.92988 


•38403 


•92332 


.40008 


.91648 


.41602 


.90036 


25 


.36812 


.92978 


•38430 


.92321 


•40035 


.91636 


.41628 


.90924 


24 


•36839 


.92967 


•38456 


•92310 


.40062 


.91625 


.41655 


.90911 


23 


.36867 


•92956 


•38483 


.92299 


.40088 


.91613 


.41681 


.90899 


22 


.36894 


•92945 


•38510 


.92287 


.40115 


.91601 


.41707 


.90887 


21 


.36921 


•92935 


•38537 


.92276 


.40141 


.91590 


.41734 


.90875 


20 


.36948 


.92924 


.38564 


.92265 


.40168 


•91578 


.41760 


.90863 


19 


.36975 


.92913 


.38591 


•92254 


.40195 


.91566 


.41787 


.90851 


18 


.37002 


.92902 


.38617 


.92243 


.40221 


•91555 


.41813 


.90839 


17 


.37029 


.92892 


.38644 


.92231 


.40248 


•91543 


.41840 


.90826 


16 


.37056 


.92881 


.38671 


.92220 


•40275 


•91531 


.41866 


.90814 


15 


.37083 


.92870 


.38698 


.92209 


•40301 


.91519 


.41892 


.90802 


14 


.37110 


.92859 


•38725 


.92198 


.40328 


.91508 


.41919 


.90790 


13 


.37137 


.92849 


.38752 


.92186 


.40355 


.91496 


.41945 


.90778 


12 


.37164 


.92838 


.38778 


.92175 


.40381 


.91484 


.41972 


.90766 


11 


.37191 


.92827 


.38805 


.92164 


.40408 


.91472 


.41998 


.90753 


10 


.37218 


.92816 


•^32 


.92152 


.40434 


.91461 


.42024 


.90741 


9 


.37245 


.92805 


.38859 


.92141 


.40461 


.91449 


.42051 


.90729 


L 


.37272 


.92794 


.38886 


.92130 


.40488 


•91437 


.42077 


.90717 


7 


.37299 


.92784 


.38912 


.92119 


.40514 


.91425 


.42104 


.90704 


6 


.37326 


•92773 


.38939 


.92107 


.40541 


.91414 


.42130 


.90692 


5 


.37353 


.92762 


.38966 


.92096 


•40567 


.91402 


.42156 


.90680 


4 


.37380 


•92751 


.38993 


.92085 


.40594 


.91390 


.42183 


.90668 


3 


.37407 


.92740 


.39020 


.92073 


.40621 


•91378 


.42209 


•90655 


2 


.37434 


.92729 


.39046 


.92062 


.40647 


.91366 


.42235 


•90643 


I 


•37461 


.92718 


.39073 


.92050 


.40674 


•91355 


.42262 


•90631 





Cosine ' 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


68 





67 


'O 


m 


>° 


6£ 


;° 





140 



MECHANICS AND ALLIED SUBJECTS 





25° 


26° 1 


27° 


28° 1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.42262 


.90631 


.43837 


.89879 


.45399 


.89101 


•46947 


.88295 


I 


.42288 


.90618 


.43863 


.89867 


.45425 


.89087 


•46973 


.88281 


2 


•42315 


.90606 


.45889 


.89854 


.45451 


.89074 


.46999 


.88267 


3 


•42341 


•90594 


.43916 


.89841 


.45477 


.89061 


.47024 


.88254 


4 


.42367 


.90582 


.43942 


.89828 


.45503 


.89048 


•47050 


.88240 


5 


.42394 


•90569 


.43968 


.89816 


•45529 


.89035 


•47076 


.88226 


6 


.42420 


•90557 


.43994 


.89803 


•45554 


.89021 


.47101 


.88213 


7 


.42446 


•90545 


.44020 


.89790 


•45580 


.89008 


.47127 


.88199 


8 


•42473 


•90532 


.44046 


•89777 


.45606 


.88995 


•47153 


.88185 


9 


.42499 


.90520 


.44072 


.89764 


•45632 


.88981 


.47178 


.88172 


lO 


.42525 


•90507 


.44098 


•89752 


•45658 


.88968 


.47204 


.88158 


II 


•42552 


.90495 


.44124 


•89739 


•45684 


.88955 


•47229 


.88144 


12 


.42578 


.90483 


•44151 


.89726 


•45710 


.88942 


•47255 


.88130 


13 


.42604 


.90470 


•44177 


•89713 


.45736 


.88928 


.47281 


.88117 


14 


.42631 


.90458 


.44203 


.89700 


.45762 


.88915 


•47306 


.88103 


IS 


.42657 


.90446 


.44229 


.89687 


•45787 


.88902 


•47332 


.88089 


i6 


.42683 


.90433 


.44255 


.89674 


.45813 


.88888 


.47358 


•88075 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


.47383 


.88062 


i8 


.42736 


.90408 


.44307 


.89649 


•45865 


.88862 


.47409 


.88048 


19 


.42762 


.90396 


•44333 


.89636 


.45891 


.88848 


.47434 


.88034 


20 


.42788 


•90383 


.44359 


.89623 


•45917 


.88835 


.47460 


.88020 


21 


.42815 


•90371 


.44385 


.89610 


.45942 


.88822 


.47486 


.88006 


22 


.42841 


•90358 


.44411 


•89597 


.45968 


.88808 


.47511 


.87993 


23 


.42867 


•90346 


.44437 


.89584 


•45994 


.88795 


.47537 


•87979 


24 


.42894 


•90334 


•44464 


.89571 


.46020 


.88782 


.47562 


•87965 


25 


.42920 


•90321 


.44490 


.89558 


.46046 


.88768 


.47588 


•87951 


26 


.42946 


.90309 


.44516 


.89545 


.46072 


.88755 


.47614 


•87937 


27 


.42972 


.90296 


•44542 


.89532 


.46097 


.88741 


.47639 


•87923 


28 


.42999 


.90284 


•44568 


.89519 


.46123 


.88728 


.47665 


.87909 


29 


.43025 


.90271 


•44594 


.89506 


.46149 


.88715 


.47690 


.87896 


30 


.43051 


•90259 


.44620 


89493 


.46175 


.88701 


.47716 


.87882 


31 


.43077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


32 


.43104 


•90233 


.44672 


.89^67 


.46226 


.88674 


.47767 


•87854 


33 


.43130 


.90221 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


34 


.43156 • 


.90208 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


35 


.43182 


.90196 


•44750 


.89428 


•46304 


.88634 


.47844 


.87812 


36 


.43209 


.90183 


•44776 


.89415 


•46330 


.88620 


.47869 


•87798 


37 


.43235 


.90171 


.44802 


.89402 


•46355 


.88607 


.47895 


.87784 


38 


.43261 


.90158 


.44828 


.89389 


•46381 


.88593 


.47920 


•87770 


39 


.43287 


.90146 


.44854 


^9376 


.46407 


.88580 


.47946 


•87756 


40 


•43313 


•90133 


.44880 


.89363 


•46433 


.88566 


.47971 


.87743 


41 


.43340 


.90120 


.44906 


.89350 


•46458 


•88553 


.47997 


•87729 


42 


.43366 


.90108 


.44932 


.89337 


.46484 


•88539 


.48022 


.87715 


43 


•43392 


•90095 


.44958 


.89324 


.46510 


.88526 


.48048 


.87701 


44 


.43418 


.90082 


•44984 


.89311 


.46536 


.88512 


•48073 


.87687 


45 


.43445 


.90070 


.45010 


.89298 


.46561 


.88499 


.48099 


.87673 


46 


.43471 


.90057 


•45036 


.89285 


•46587 


.88485 


.48124 


•87659 


=57 


.43497 


•90045 


.45062 


.89272 


•46613 


.88472 


.48150 


•87645 


^8 


.43523 


.90032 


.45088 


.89259 


.46639 


.88458 


.48175 


•87631 


49 


.43549 


.90019 


.45114 


.89245 


.46664 


•88445 


.48201 


.87617 


50 


.43575 


.90007 


.45140 


.89232 


.46690 


.88431 


.48226 


•87603 


51 


.43602 


•89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


52 


.43628 


.89981 


45192 


.89206 


.46742 


.88404 


•48277 


•87575 


53 


•43654 


.89968 


.45218 


.89193 


.46767 


•88390 


•48303 


•87561 


54 


.43680 


.89956 


.45243 


.89180 


•46793 


•88377 


•48328 


•87546 


55 


.43706 


•89943 


.45269 


.89167 


.46819 


•88363 


•48354 


•87532 


56 


.43733 


•89930 


.45295 


.89153 


.46844 


.88349 


.48379 


.87518 


57 


.43759 


.89918 


.45321 


.89140 


.46870 


.88336 


.48405 


•87504 


58 


.43785 


.89905 


.45347 


.89127 


.46896 


.88322 


.48430 


.87490 


59 


.43811 


.89892 


.45373 


.89114 


.46921 


.88308 


.48456 


•87476 


60 


•43837 


.89870 


•45399 


.89101 


.46947 


•88295 


.48481 


.87462 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 




6^ 


t° I 


6c 


J° 


6^ 


>o 

•J 


6] 


L° ) 



MEASUREMENT OF RIGHT TRIANGLES 



141 





29<^ 1 


30° 1 


31° 


32° 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


f 


o 


.48481 


.87462 


.50000 


.86603 


.51504 


.85717 


.52992 


.84805 


60 


I 


.48506 


.87448 


.50025 


.86588 


.51529 


•85702 


•53017 


.84789 


5g 


2 


.48532 


.87434 


■50050 


•86573 


.51554 


.85687 


•53041 


.84774 


58 


3 


.48557 


.87420 


.50076 


.86559 


.51579 


.85672 


.53066 


.84759 


57 


4 


.48583 


.87406 


.50101 


•86544 


.51604 


.85657 


.53091 


•84743 


56 


5 


.48608 


.87391 


.50126 


.86530 


.51628 


.85642 


.53115 


.84728 


55 


6 


.48634 


•87377 


.50151 


.86515 


.51653 


.85627 


•53140 


.84712 


54 


7 


.48659 


•87363 


•50176 


.86501 


.51678 


.85612 


.53164 


.84697 


53 


8 


.48684 


.87349 


.50201 


.86486 


•51703 


•85597 


.53189 


.84681 


52 


9 


.48710 


•87335 


.50227 


.86471 


•51728 


.85582 


.53214 


.84666 


51 


lO 


.48735 


.87321 


•50252 


.86457 


•51753 


.85567 


.53238 


.84650 


50 


II 


.48761 


•87306 


.50277 


.86442 


•51778 


.85551 


.53263 


•84635 


49 


12 


.48786 


.87292 


.50302 


.86427 


.51803 


.85536 


.53288 


.84619 


48 


13 


.48811 


.87278 


•50327 


.86413 


.51828 


.85521 


.53312 


.84604 


47 


14 


.48837 


.87264 


•50352 


.86398 


.51852 


.85506 


.53337 


.84588 


46 


IS 


.48862 


.87250 


•50377 


.86384 


•51877 


.85491 


.53361 


•84573 


45 


i6 


.48888 


•87235 


.50403 


.86369 


.51902 


.85476 


.53386 


•84557 


44 


17 


.48913 


.87221 


.50428 


.86354 


•51927 


.85461 


.53411 


•84542 


43 


i8 


.48938 


.87207 


.50453 


.86340 


•51952 


.85446 


.53435 


•84526 


42 


19 


.48964 


.87193 


.50478 


•86325 


•51977 


.85431 


.53460 


•845 II 


41 


20 


.48989 


.87178 


.50503 


.86310 


.52002 


.85416 


.53484 


.84495 


40 


21 


.49014 


.87164 


.50528 


.86295 


.52026 


.85401 


.53509 


.84480 


39 


22 


.49040 


.87150 


.50553 


.86281 


.52051 


.85385 


.53534 


.84464 


38 


23 


.49065 


.87136 


.50578 


.86266 


.52076 


.85370 


•53558 


.84448 


37 


24 


.49090 


.87121 


.50603 


.86251 


.52101 


.85355 


.53583 


.84433 


36 


25 


.49116 


.87107 


.50628 


•86237 


.52126 


.85340 


.53607 


.84417 


35 


26 


.49141 


.87093 


•50654 


.86222 


.52151 


•85325 


•53632 


.84402 


34 


27 


.49166 


.87079 


•50679 


.86207 


•52175 


.85310 


.53656 


.84386 


33 


28 


.49192 


.87064 


.50704 


.86192 


.52200 


.85294 


.53681 


.84370 


32 


29 


.49217 


.87050 


.50729 


.86178 


.52225 


.85279 


.53705 


•84355 


31 


30 


.49242 


.87036 


.50754 


.86163 


^52250 


.85264 


.53730 


•84339 


30 


31 


.49268 


.87021 


.50779 


.86148 


•52275 


.85249 


.53754 


.84324 


29 


32 


.49293 


.87007 


.50804 


.86133 


•52299 


.85234 


.53779 


.84308 


28 


33 


.49318 


.86993 


.50829 


.86119 


•52324 


.85218 


.53804 


.84292 


27 


34 


.49344 


.86978 


.50854 


.86104 


•52349 


.85203 


.53828 


•84277 


26 


35 


.49369 


.86964 


.50879 


.86089 


•52374 


.85188 


.53853 


.84261 


25 


36 


.49394 


.86949 


•50904 


.86074 


•52399 


.85173 


.53877 


.84245 


24 


37 


.49419 


.86935 


.50929 


.86059 


.52423 


.85157 


.53902 


.84230 


23 


38 


.49445 


.86921 


•50954 


.86045 


.52448 


.85142 


•53926 


.84214 


22 


39 


.49470 


.86906 


•50979 


.86030 


•52473 . 


.85127 


•53951 


.84198 


21 


40 


.49495 


.86892 


.51004 


.86015 


.52498 


.85112 


.53975 


.84182 


20 


41 


.49521 


.86878 


•51029 


.86000 


.52522 


.85096 


.54000 


.84167 


19 


42 


•49546 


.86863 


•51054 


.85985 


.52547 


.85081 


.54024 


.84151 


18 


43 


.49571 


.86849 


•51079 


.85970 


.52572 


.85066 


.54049 


.84135 


17 


44 


.49596 


.86834 


•51104 


.85956 


.52597 


.85051 


.54073 


.84120 


16 


45 


.49622 


.86820 


•51129 


.85941 


.52621 


.85035 


.54097 


.84104 


15 


46 


.49647 


.86S05 


•51154 


.85926 


.52646 


.85020 


•54122 


.84088 


14 


47 


.49672 


.86791 


•51179 


.85911 


.52671 


.85005 


.54146 


.84072 


13 


48 


.49697 


.86777 


.51204 


.85896 


.52696 


.84989 


•54171 


.84057 


12 


49 


.49723 


.86762 


.51229 


.85881 


.52720 


.84974 


•54195 


.84041 


11 


50 


.49748 


.86748 


.51254 


.85866 


.52745 


.84959 


.54220 


■84025 


10 


51 


.49773 


•86733 


•51279 


.85851 


.52770 


.84943 


.54244 


.84009 


9 


52 


.49798 


.86719 


•51304 


.85836 


.52794 


.84928 


.54269 


.83994 


8 


53 


.49824 


.86704 


•51329 


.85821 


.52819 


.84913 


.54293 


•83978 


7 


54 


.49849 


.86690 


.51354 


.85806 


.52844 


.84897 


.54317 


.83962 


6 


55 


.49874 


.86675 


•51379 


.85792 


.52869 


.84882 


.54342 


.83946 


5 


56 


.49899 


.86661 


.51404 


.85777 


.52893 


.84866 


.54366 


•83930 


4 


57 


.49924 


.86646 


•51429 


.85762 


.52918 


.84851 


.54391 


•83915 


3 


58 


•49950 


.86632 


•51454 


.85747 


.52943 


.84836 


.54415 


•83899 


2 


59 


•49975 


.86617 


•51479 


.85732 


.52967 


.84820 


.54440 


.83883 


I 


60 


.50000 


.86603 


•51504 


.85717 


.52992 


.84805 


.54464 


•83867 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




6( 


r 


5< 


r 


5i 


^° 


5' 


r 





142 



MECHANICS AND ALLIED SUBJECTS 





33° 


34° 


35° 


36° 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


COSNIE 


e 


o 


.54464 


.83867 


•55919 


.82904 


.57358 


.81915 


.58779 


.80902 


60 


I 


.54488 


.83851 


•55943 


.82887 


.57381 


.81899 


.58802 


.80885 


59 


2 


.54513 


.83835 


•55968 


.82871 


.57405 


.81882 


.58826 


.80867 


58 


3 


•54537 


.83819 


•55992 


.82855 


.57429 


.81865 


.58849 


.80850 


57 


4 


.54561 


.83804 


.56016 


.82839 


.57453 


.81848 


.58873 


.80833 


56 


5 


.54586 


.83788 


.56040 


.82822 


.57477 


.81832 


.58896 


.80816 


55 


6 


.54610 


.83772 


.56064 


.82806 


.57501 


.81815 


.58920 


.80799 


54 


7 


•54635 


.83756 


.56088 


.82790 


.57524 


.81798 


.58943 


.80782 


53 


8 


.54659 


.83740 


.56112 


.82773 


.57548 


.81782 


.58967 


.80765 


52 


9 


.54683 


.83724 


•56136 


.82757 


.57572 


.81765 


.58990 


^748 


51 


lO 


.54708 


.83708 


.56160 


.82741 


.57596 


.81748 


.59014 


.80730 


50 


II 


.54732 


83692 


.56184 


.82724 


.57619 


.81731 


.59037 


.80713 


49 


12 


.54756 


.83676 


.56208 


.82708 


.57643 


.81714 


.59061 


.80696 


48 


13 


•54781 


.83660 


.56232 


.82692 


.57667 


.81698 


.59084 


.80679 


47 


14 


.54805 


.83645 


.56256 


.82675 


.57691 


.81681 


.59108 


.80662 


46 


15 


.54829 


.83629 


.56280 


.82659 


.57715 


.81664 


.59131 


.80644 


45 


16 


.54854 


.83613 


.56305 


.82643 


.57738 


.81647 


.59154 


.80627 


44 


r? 


.54878 


.83597 


•56329 


.82626 


.57762 


.81631 


.59178 


.80610 


43 


18 


.54902 


.83581 


•56353 


.82610 


.57786 


.81614 


.59201 


.80593 


42 


19 


.54927 


.83565 


•56377 


.82593 


.57810 


.81597 


.59225 


.80576 


41 


20 


.54951 


•83549 


.56401 


.82577 


.57833 


.81580 


.59248 


.8o55« 


40 


21 


.54975 


•83533 


.56425 


.82561 


.57857 


.81563 


.59272 


.80541 


39 


22 


.54999 


•83517 


.56449 


.82544 


.57881 


.81546 


.59295 


.80524 


38 


23 


.55024 


.83501 


.56473 


.82528 


•57904 


.81530 


.59318 


.80507 


37 


24 


•55048 


•83485 


.56497 


.82511 


.57928 


.81513 


•59342 


.80489 


36 


25 


.55072 


.83469 


.56521 


.82495 


.57952 


.81496 


.59365 


.80472 


35 


26 


.55097 


.83453 


•56545 


.82478 


.57976 


.81479 


•59389 


.80455 


34 


27 


.55121 


.83437 


•56569 


.82462 


.57999 


.81462 


.59412 


.80438 


33 


28 


.55145 


.83421 


•56593 


.82446 


.58023 


.81445 


•59436 


.80420 


32 


29 


•55169 


.83405 


.56617 


.82429 


.58047 


.81428 


•59459 


.80403 


31 


30 


.55194 


.83389 


•56641 


.82413 


.58070 


.81412 


.59482 


.80386 


30 


31 


.55218 


.83373 


•56665 


.82396 


.58094 


.81395 


•59506 


.80368 


29 


32 


.55242 


.83356 


.56689 


.82340 


.58118 


.81378 


•59529 


.80351 


28 


33 


.55266 


.83340 


.56713 


.82363 


.58141 


.81361 


•59552 


.80334 


27 


34 


.55291 


.83324 


.56736 


.82347 


.58165 


.81344 


•59576 


.80316 


26 


35 


.55315 


•83308 


.56760 


.82330 


.58189 


.81327 


•59599 


.80299 


25 


36 


.55339 


.83292 


.56784 


.82314 


.58212 


.81310 


.59622 


.80282 


24 


37 


.55363 


•83276 


.56808 


.82297 


.58236 


.81293 


•59646 


.80264 


23 


38 


.55388 


.83260 


.56832 


.82281 


.58260 


.81176 


•59669 


.80247 


22 


39 


.55412 


.83244 


.56856 


.82264 


.58283 


.81259 


•59693 


.80230 


21 


40 


.55436 


.83228 


.56880 


.82248 


.58307 


.81242 


•59716 


.80212 


20 


41 


.55460 


.83212 


.56904 


.82231 


.58330 


.81225 


.59739 


.80195 


19 


42 


.55484 


•83195 


.56928 


.82214 


.58354 


.81208 


.59763 


.80178 


18 


43 


.55509 


.83179 


.56952 


.82198 


.58378 


.81191 


•59786 


.80160 


17 


44 


.55533 


.83163 


.56976 


.82181 


.58401 


.81174 


.59809 


.80143 


16 


45 


•55557 


•83147 


.57000 


.82165 


.58425 


.81157 


.59832 


.80125 


15 


46 


•55581 


•83131 


•57024 


.82148 


.58449 


.81140 


.59856 


.80108 


14 


47 


.55605 


.83115 


.57047 


.82132 


.58472 


.81123 


.59879 


.80091 


13 


48 


.55630 


.83098 


.57071 


.82115 


.58496 


.81106 


•59902 


.80073 


12 


49 


.55654 


.83082 


•57095 


.82098 


.58519 


.81089 


.59926 


.80056 


II 


50 


.55678 


.83066 


.57119 


.82082 


.58543 


.81072 


.59949 


.80038 


10 


51 


.55702 


.83050 


•57143 


.82065 


.58567 


.81055 


.59972 


.80021 


9 


52 


.55726 


•83034 


•57167 


.82048 


.58590 


.81038 


.59995 


.80003 


8 


53 


.55750 


.83017 


.57191 


.82032 


.58614 


.81021 


.60019 


.79986 


7 


54 


.55775 


.83001 


.57215 


.82015 


.58637 


.81004 


.60042 


.79968 


6 


55 


.55799 


.82985 


.57238 


.81999 


.58661 


.80987 


.60065 


.79951 


5 


56 


.55823 


.82969 


.57262 


.81982 


.58684 


.80970 


.60089 


.79934 


4 


57 


•55847 


•82953 


.57286 


.81965 


.58708 


.80953 I 


.60112 


.79916 


3 


58 


.55871 


.82936 


.57310 


.81949 


.58731 


.80936 


.60135 


.79899 


2 


1^ 


•55895 


.82920 


.57334 


.81932 


.58755 


.80919 


.60158 


.79881 


I 


5o 


.55919 


.82904 


.57358 


.81915 


.58779 


.80902 


.60182 


.79864 





/ 


Cosine 


Sine 


Cosine I 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




5€ 


►° 1 


55 


>° 


54 


0. 


52 


►° 





MEASUREMENT OF RIGHT TRIANGLES 



143 



37° 


38° 


39° 


40° 




Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 Cosine 


/ 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 


60 


.60205 


.79846 


.61589 


.78783 


.62955 


.77696 


.64301 


.76586 


59 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


•64323 


•76567 


58 


.60251 


.79811 


.61635 


•78747 


.63000 


.77660 


.64346 


•76548 


57 


.60274 


.79793 


.61658 


.78729 


.63022 


.77641 


.64368 


•76530 


56 


.60298 


.79776 


.61681 


.78711 


•63045 


.77623 


.64390 


.76511 


55 


.60321 


.79758 


.61704 


.78694 


.63068 


.77605 


.64412 


.76492 


54 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


.64435 


.76473 


53 


.60367 


.79723 


•61749 


.78658 


•631 13 


.77568 


.64457 


.76455 


52 


.60390 


.79706 


.61772 


.78640 


•63135 


•77550 


.64479 


.76436 


51 


.60414 


.79688 


.61795 


.78622 


.63158 


•77531 


.64501 


.76417 


50 


.60437 


.79671 


.61818 


.78604 


.63180 


•77513 


.64524 


.76398 


49 


.60460 


.79653 


.61841 


.78586 


•63203 


•77494 


.64546 


.76380 


48 


.60483 


.79635 


.61864 


.78568 


•63225 


'77476 


.64568 


.76361 


47 


.60506 


.79618 


.61887 


•78550 


.63248 


.77458 


.64590 


.76342 


46 


.60529 


.79600 


.61909 


•78532 


•63271 


.77439 


.64612 


.76323 


45 


.60553 


.79583 


.61932 


•78514 


.63293 


.77421 


.64635 


.76304 


44 


.60576 


.79565 


.61955 


.78496 


.63316 


.77402 


•64657 


.76286 


43 


.60599 


.79547 


.61378 


.78478 


.63338 


.77384 


.64679 


.76267 


42 


.60622 


.79530 


.62001 


.78460 


.63361 


.77366 


.64701 


.76248 


41 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


.64723 


.76229 


40 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


.64746 


.76210 


39 


.60691 


.79477 


.62069 


•78405 


.63428 


•77310 


.64768 


.76192 


38 


.60714 


•79459 


.62092 


•78387 


.63451 


.77292 


.64790 


.76173 


37 


.60738 


.79441 


.62115 


•78369 


•63473 


.77273 


.64812 


.76154 


36 


.60761 


.79424 


.62138 


•78351 


.63496 


•77255 


.64834 


.76135 


35 


.60784 


.79406 


.62160 


•78333 


.63518 


.77236 


.64856 


.76116 


34 


.60807 


.79388 


.62183 


.78315 


.63540 


.77218 


.64878 


.76097 


33 


.60830 


.79371 


.62206 


•78297 


•63563 


.77199 


.64901 


.76078 


3a 


.60853 


.79353 


.62229 


.78279 


•63585 


.77181 


.64923 


•76059 


31 


.60876 


•79335 


.62251 


.78261 


.63608 


.77162 


.64945 


.76041 


30 


.60899 


•79318 


.62274 


•78243 


.63630 


.77144 


.64967 


.76022 


29 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


.64989 


.76003 


28 


.60945 


.79282 


.62320 


.78206 


.63675 


•77107 


.65011 


.75984 


27 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


.65033 


.75965 


26 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


•65055 


.75946 


25 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


•65077 


.75927 


24 


.61038 


.79211 


.62411 


•78134 


.63765 


.77033 


.65100 


•75908 


23 


.61061 


.79193 


•62433 


.78116 


.63787 


.77014 


.65122 


•75889 


22 


.61084 


.79176 


.62456 


.78098 


.63810 


.76996 


.65144 


•75870 


21 


.61107 


•79158 


.62479 


.78079 


.(i3^32 


•76977 


.65166 


•75851 


20 


.61130 


.79140 


.62502 


.78061 


.63854 


•76959 


.65188 


•75832 


19 


.61153 


.79122 


.62524 


.78043 


.63877 


.76940 


.65210 


.75813 


18 


.61176 


.79105 


.62547 


.78025 


.63899 


.76921 


.65232 


.75794 


17 


.61199 


.79087 


.62570 


.78007 


.63922 


•76903 


.65254 


.75775 


16 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


.65276 


.75756 


15 


.61245 


.79051 


.62615 


•77970 


.63966 


.76866 


.65298 


.75738 


14 


.61268 


.79033 


.62638 


.77952 


.63989 


•76847 


.65320 


•75719 


13 


.61291 


•79016 


.62660 


.77934 


.64011 


.76828 


.65342 


.75700 


12 


.61314 


.78098 


.62683 


.77916 


.64033 


.76810 


.65364 


.75680 


II 


•61337 


.78980 


.62706 


•77897 


.64056 


.76791 


.65386 


.75661 


10 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


.65408 


.75642 


9 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


.65430 


.75623 


8 


.61406 


.78926 


•62774 


•77843 


.64123 


.76735 


.65452 


.75604 


7 


.61429 




.62796 


.77824 


.64145 


.76717 


•65474 


.75585 


6 


.61451 


.78891 


.62819 


.77806 


.64167 


.76698 


.65496 


•75566 


5 


.61474 


.78873 


.62842 


•77788 


.64190 


.76679 


.65518 


•75547 


4 


.61497 


.78855 


.62864 


•77769 


.64212 


.76661 


.65540 


•75528 


3 


.61520 


.78837 


.62887 


•77751 


.64234 


.76642 


.65562 


•75509 


2 


•61543 


.78819 


.62909 


•77733 


.64256 


•76623 i 


.65584 


.75490 


I 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 


.65606 


.75471 





Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 


52 





51 





50 


1 


49 








44 


^ 


lECHu 


iNICS 


AND 


ALLIED SUBJECTS 




41° 


42° 


43° 


440 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 Cosine 


Sine 


Cosine 


O 


.65606 


.75471 


.66913 


.74314 


.68200 


•73135 


.69466 


•71934 


I 


.65628 


.75452 


.66935 


.74295 


.68221 


.73116 


.69487 


.71914 


2 


.65650 


.75433 


.66956 


.74276 


.68242 


.73096 


.69508 


.71894 


3 


.65672 


.75414 


.669,-8 


.74256 


.68264 


•73076 


.69529 


.71873 


4 


.65694 


.75395 


.66999 


•74227 


.68285 


.73056 


•69549 


.71853 


, 5 


.65716 


.75375 


.67021 


.74217 


.68306 


.73036 


.69570 


•71833 


6 


.65738 


•75356 


.67043 


•74198 


.68327 


.73016 


.69591 


.71813 


7 


.65759 


.75337 


.67064 


•74178 


.68349 


.72996 


.69612 


.71792 


8 


.65781 


•75318 


.67086 


•7JI59 


.68370 


•72J76 


•69633 


.71772 


9 


.65803 


.75299 


.67107 


•74139 


.68391 


.72957 


.69654 


•71752 


lO 


.65825 


.75280 


.67129 


•74120 


.68412 


•72937 


•69675 


.71732 


II 


.65847 


.75261 


.67151 


.74100 


•68434 


.72917 


.69696 


.71711 


12 


.65869 


•75241 


.67172 


.74080 


.68455 


•72897 


.69717 


.71691 


13 


.65891 


.75222 


.67194 


.74061 


.68476 


.72377 


.69737 


.71671 


14 


.65913 


•75203 


.67215 


.74041 


.68497 


•72857 


.69758 


.71650 


IS 


.65935 


•75184 


.67237 


.74022 


.68518 


•72837 


.69779 


.71630 


i6 


.65956 


.75165 


.67258 


./4002 


•68539 


.72817 


.69800 


.71610 


17 


.65978 


.75146 


.67280 


.73983 


.68561 


•72797 


.69821 


.71590 


i8 


.66000 


.75126 


.67301 


•73963 


.68582 


.72777 


.69842 


.71569 


19 


.66022 


.75107 


.67323 


•73944 


.68603 


.72757 


.69862 


.71549 


20 


.66044 


.75088 


.67344 


•73924 


.68624 


•72737 


.69883 


.71529 


21 


.66066 


.75069 


.67366 


.73904 


.68645 


.72717 


.69904 


.71508 


22 


.66088 


•75050 


.67387 


.73885 


.68666 


.72697 


.69925 


.71488 


23 


.66109 


.75030 


.67409 


.73865 


.68688 


•72677 


.69946 


.71468 


24 


.66131 


.75011 


.67430 


.73846 


.68709 


•72657 


.69966 


.71447 


25 


.66153 


.74992 


•67452 


.73826 


.68730 


.72637 


.69987 


.71427 


26 


.66175 


.74973 


•67473 


.73806 


.68751 


.72617 


.70008 


.71407 


27 


.66197 


•74953 


•67495 


•73787 


.68772 


•72597 


.70029 


.71386 


28 


.66218 


.74934 


.67516 


•73767 


.68793 


.72577 


.70049 


.71366 


2? 


.66240 


.74915 


.67538 


.73747 


.68814 


.72557 


.70070 


.71345 


30 


.66262 


.74896 


.67559 


•73728 


•68835 


•72537 


.70091 


.71325 


31 


.66284 


.74876 


.67580 


.73708 


.68857 


•72517 


.70112 


.71305 


32 


,66306 


.74857 


.67602 


.73688 ; 


.68878 


.72497 


.70132 


.71284 


33 


.66327 


•74838 


.67623 


•73669 ; 


.68899 


•72477 


.70153 


.71264 


34 


.66349 


.74818 


.67645 


•73649 i 


.68920 


•72457 


.70174 


.71243 


35 


.66371 


.74799 


.67666 


.73629 


.68941 


•72437 


•70195 


.71223 


36 


.66393 


.74780 


.67688 


•73610 


.68962 


.72417 


.70215 


.71203 


37 


.66414 


.74760 


.67709 


.73590 


.68983 


.72397 


.70236 


.71182 


38 


.66436 


•74741 


.67730 


•73570 


.69004 


.72377 


•70257 


.71162 


39 


.66458 


•74722 


•67752 


•73551 


.69025 


•72357 


.70277 


.71141 


40 


.66480 


•74703 


•67773 


•73531 


.69046 


.72337 


.70298 


.71121 


41 


.66501 


.74683 


.67795 


•73511 


.69067 


.72317 


•70319 


.71100 


42 


.66523 


.74664 


.67816 


.73491 


.69088 


.72297 


•70339 


.71080 


43 


•66545 


.74644 


.67837 


.73472 


.69109 


.72277 


.70360 


•71059 


44 


.66566 


.74625 


.67859 


.73452 


.69130 


.72257 


.70381 


•71039 


45 


.66588 


.74606 


.67880 


.73432 


.69151 


.72236 


.70401 


.71019 


46 


.66610 


•74586 


.67901 


•73413 


.69172 


.72216 


.70422 


.70998 


47 


.66632 


•74567 


•67923 


•73393 


.69193 


.72196 


•70443 


.70978 


48 


.66653 


•74548 


.67944 


•73373 


.69214 


.72176 


.70463 


•70957 


49 


.66675 


•74528 


•67965 


•73353 


•69235 


.72156 


.70484 


•70937 


50 


.66697 


•74509 


•67987 


•73333 


.69256 


.72136 


.70505 


.70916 


51 


.66718 


.74489 


.68008 


.73314 


.69277 


.72116 


.70525 


.70896 


52 


.66740 


.74470 .68029 1 


•73294 


.69298 


•72095 


•70546 


.70875 


53 


.66762 


.74451 


.68051 


.73274 


.69319 


•72075 


•70567 


.70855 


54 


.66783 


•74431 


.68072 


•73254 


.69340 


•72055 


.70587 


.70834 


55 


.66805 


•74412 


.68093 


•73234 


.69361 


•72035 


.70608 


.70813 


56 


.66827 


•74392 


.68115 


•73215 


.69382 


.72015 


.70628 


.70793 


57 


.66848 


.74373 


.68136 


.73195 


.69403 


•71995 


.70649 


.70772 


58 


.66870 


.74353 


.68157 


•73175 


.69424 


.71974 


.70670 


.70752 


59 


.66891 


.74334 


.68179 


•73155 


.69445 


.71954 


.70690 


.70731 


5o 


.66913 


•74314 


.68200 


•7313s 


.69466 


.71934 1 


.70711 


.70711 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 




48 


° 


47 


'O 


4e 


>° 


45 






MEASUREMENT OF RIGHT TRIANGLES 



145 



c 


)° 


1 





2° I 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


.ocx>oo 


Infinite. 


.01746 


57.2900 


.03492 


28.6363 


.00029 


3437-750 


.01775 


56.3506 


.03521 


28.3994 


.00058 


1718.870 


.01804 


55-4415 


.03550 


28.1664 


.00087 


1145.920 


.01833 


54-5613 


.03579 


27-9372 


.00116 


859-436 


.01862 


53.7086 


.03609 


27.7117 


.00145 


687.549 


.01891 


52.8821 


.03638 


27.4899 


.00175 


572.957 


.01920 


52.0807 


.03667 


27-2715 


.00204 


491.106 


.01949 


51-3032 


.03696 


27.0566 


.00233 


429.718 


.01978 


50.5485 


.03725 


26.8450 


.00262 


381.971 


.02007 


49-8157 


•03754 


26.6367 


.00291 


343-774 


.02036 


49.1039 


•03783 


26.4316 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


.00349 


286.478 


.02095 


47-7395 


.03842 


26.0307 


.00378 


264.441 


.02124 


47-0853 


.03871 


25-8348 


.00407 


245-552 


.02153 


46.4489 


.03900 


25-6418 


.00436 


229.182 


.02182 


45-8294 


.03929 


25.4517 


.00465 


214.858 


.02211 


45.2261 


.03958 


25.2644 


.00495 


202.219 


.02240 


44.6386 


.03987 


25.0798 


.00524 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.00553 


180.932 


.02298 


43.5081 


.04046 


24.7185 


.00582 


171-885 


.02328 


42.9641 


.04075 


24.5418 


.00611 


163.700 


.02357 


42-4335 


.04104 


24-3675 


.00640 


156-259 


.02386 


41.9158 


•04133 


24.1957 


.00669 


149.465 


.02415 


41.4106 


.04162 


24.0263 


.00698 


143-237 


.02444 


40.9174 


.04191 


23-8593 


.00727 


137-507 


.02473 


40.4358 


.04220 


23-6945 


.00756 


132.219 


.02502 


39-9655 


.04250 


23-5321 


.00785 


127-321 


.02531 


39.5059 


.04279 


23-3718 


.00814 


122.774 


.02560 


39.0568 


.04308 


23.2137 


.00844 


118.540 


.02589 


38.6177 


•04337 


23-0577 


.00873 


114-589 


.02619 


38.1885 


.04366 


22.9038 


.00902 


110.892 


.02648 


37.7686 


•04395 


22.7519 


.00931 


107.426 


.02677 


37-3579 


.04424 


22.6020 


.00960 


104. 171 


.02706 


36.9560 


.04454 


22.4541 


.00989 


101.107 


.02735 


36.5627 


.04483 


22.3081 


.01018 


98.2179 


.02764 


36.1776 


.04512 


22.1640 


.01047 


95-4895 


.02793 


35-8006 


.04541 


22.0217 


.01076 


92.9085 


.02822 


35-4313 


•04570 


21.8813 


.01105 


90.4633 


.02851 


35-0695 


.04599 


21.7426 


•01 1 35 


88.1436 


.02881 


34-7151 


.04628 


21.6056 


.01164 


85-9398 


.02910 


34-3678 


.04658 


21.4704 


.01193 


83-8435 


.02939 


34-0273 


.04687 


21.3369 


.01222 


81.8470 


.02968 


33-6935 


.04716 


21.2049 


.01251 


79-9434 


.02997 


33.3662 


•04745 


21.0747 


.01280 


7S.1263 


.03026 


33-0452 


.04774 


20.9460 


.01309 


76.3900 


.03055 


32.7303 


.04803 


20.8188 


.01338 


74-7292 


.03084 


32.4213 


.04832 


20.6932 


.01367 


73.1390 


.03114 


32.1181 


.04862 


20.5691 


.01396 


71.6151 


-03143 


31.8205 


.04891 


20.4465 


.01425 


70.1533 


.03172 


31.5284 


.04920 


20.3253 


•01455 


68.7501 


.03201 


31.2416 


-04949 


20.2056 


.01484 


67.4019 


.03230 


30.9599 


.04978 


20.0872 


•01513 


66.1055 


.03259 


30.6833 


.05007 


19.9702 


.01542 


64.8580 


.03288 


30.4116 


•05037 


19.8546 


.01571 


63.6567 


-03317 


30.1446 


.05066 


19.7403 


.01600 


62.4992 


•03346 


29.8823 


•05095 


19-6273 


.01629 


61.3829 


•03376 


29.6245 


.05124 


19-5156 


.01658 


60.3058 


.03405 


29.3711 


.05153 


19-4051 


.01687 


59.2659 


.03434 


29.1220 


.05182 


19.2959 


.01716 


58.2612 


.03463 


28.8771 


.05212 


19.1879 


.01746 


57-2900 


-03492 


28.6363 


.05241 


19.081 1 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


^ 


19° 


8 


8° 


8 


70 



3° 
Tan. Co-tan. 



.05241 
.05270 
.05299 
.05328 
•05357 
.05387 
.05416 

•05445 
.05474 
.05503 
.05533 
.05562 

.05591 
005620 

.05649 
.05678 
.05708 
.05737 
.05766 

.05795 
.05824 

•05854 
.05883 
.05912 
.05941 
.05970 

•05999 
.06029 
.06058 
.06087 
.06116 
•06145 
.06175 
.06204 
.06233 
.06262 
.06291 
.06321 
•06350 
•06379 
.06408 

•06437 
.06467 
.06496 
.06525 

.06554 
.06584 
.06613 
.06642 
.06671 
.06700 

.06730 
.06759 
.06788 
.06817 
.06847 
.06876 
.06905 
.06934 
.06963 
•06993 



9.081 1 
8.9755 
8.871 1 
8.7678 
8.6656 

8.564s 
8.4645 

8.3655 
8.2677 
8.1708 
8.0750 

7.9802 
7-8863 
7-7934 
7-7015 
7.6106 
7^5205 
7-4314 
7-3432 
7-2558 
7.1693 

7-0837 
6.9990 
6.9150 
6.8319 
6.7496 
6.6681 
6.5874 
6.5075 
6.4283 
6.3499 

6.2722 
6.1952 
6.1 190 
6.0435 
5.9687 
5.8945 
5.8211 

5.7483 
5.6762 
5.6048 

5-5340 
5-4638 
5-3943 
5-3254 
5-2571 
5-1893 
5.1222 
5.0557 
4.9898 
4.9244 

4.8596 
4-7954 
4-7317 
4.668s 
4-6059 
4.5438 
4-4823 
4.4212 
4-3607 
4-3007 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 

45 

44 
43 
42 
41 
40 

39 
38 
3/ 
36 
35 
34 
33 
32 
31 
30 

29 
28 

27 
26 

25 
24 

23 
22 
21 

20 

15 
18 
17 
16 

15 
14 
13 
12 
II 
10 

9 

8 
7 
6 

5 
4 
3 
2 

I 



Co-tan. Tan. 
86° 



10 



146 



MECHANICS AND ALLIED SUBJECTS 





4^ 


3 


5 





6 





7 





/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


o 


.06993 


14-3007 


.08749 


1 1. 4301 


.10510 


9-51436 


.12278 


8.14435 


I 


.07022 


14.2411 


.08778 


II.3919 


.10540 


9.48781 


.1230S 


8.12481 


2 


.07051 


14.182I 


.08807 


11.3540 


•10569 


9.46141 


•12338 


8.10536 


3 


.07080 


14.1235 


.08837 


II.3163 


.10599 


9-43515 


•12367 


8.08600 


4 


.07110 


14-0655 


.08866 


11.2789 


.10628 


9.40904 


•12397 


8.06674 


S 


•07139 


14.0079 


.08895 


II.2417 


•10657 


9-38307 


.12426 


8.04756 


6 


.07168 


13-9507 


.08925 


11.2048 


.10687 


9-35724 


.12456 


8.02848 


7 


.07197 


13.8940 


•08954 


II.1681 


.10716 


9-33154 


.12485 


8.00948 


8 


.07227 


13-8378 


.08983 


II.I316 


.10746 


9-30599 


.12515 


7-99058 


9 


.07256 


13.7821 


•09013 


11-0954 


.10775 


9.28058 


.12544 


7.97176 


lO 


.07285 


13-7267 


.09042 


11.0594 


.10805 


9-25530 


.12574 


7.95302 


ri 


.07314 


13.6719 


.09071 


11.0237 


•10834 


9.23016 


.12603 


7.93438 


12 


.07344 


13.6174 


.09101 


10.9882 


.10863 


9.20516 


.12633 


7-91582 


13 


•07373 


13-5634 


.09130 


10.9529 


•10893 


9.18028 


.12662 


7-89734 


14 


.07402 


13.5098 


.09159 


10.9178 


.10922 


9-15554 


.12692 


7-87895 


15 


•07431 


13-4566 


.09189 


10.8829 


•10952 


9-13093 


.12722 


7.86064 


i6 


.07461 


13-4039 


.09218 


10.8483 


.10981 


9.10646 


.12751 


7.84242 


17 


.07490 


13-3515 


.09247 


10.8139 


.11011 


9.08211 


.12781 


7.82428 


i8 


•07519 


13-2996 


.09277 


10.7797 


.11040 


9.05789 


.12810 


7.80622 


19 


•07548 


13.2480 


.09306 


10.7457 


.11070 


9-03379 


.12840 


7.78825 


20 


.07578 


13-1969 


•09335 


IO.7119 


. 11099 


9-00983 


.12869 


7.77035 


21 


.07607 


13-1461 


•09365 


10.6783 


.11128 


8.98598 


.12899 


7.75254 


22 


.07636 


13.0958 


•09394 


10.6450 


.11158 


8.96227 


.12929 


7-73480 


23 


.07665 


13.0458 


.09423 


10.61 18 


.11187 


8.93867 


.12958 


7.71715 


24 


.07695 


12.9962 


•09453 


10.5789 


.11217 


8.91520 


.12988 


7-69957 


25 


.07724 


12.9469 


.09482 


10.5462 


.11246 


8.89185 


.13017 


7.68208 


26 


•07753 


12.8981 


.095 1 1 


10.5136 


.11276 


8.86862 


•13047 


7.66466 


27 


.07782 


12.8496 


.09541 


10.4813 


•11305 


8.84551 


.13076 


7-64732 


28 


.07812 


12.8014 


.09570 


10.4491 


•11335 


8.82252 


.13106 


7.6300s 


29 


.07841 


12.7536 


.09600 


10.4172 


•11364 


8.79964 


.13136 


7.61287 


30 


.07870 


12.7062 


.09629 


10.3854 


•11394 


8.77689 


.13165 


7-59575 


31 


.07899 


12.6591 


.09658 


10.3538 


.11423 


8.75425 


•13195 


7.57872 


32 


.07929 


12.6124 


.09688 


10.3224 


.11452 


8.73172 


•13224 


7.56176 


33 


.07958 


12.5660 


.09717 


10.2913 


.11482 


8.70931 


•13254 


7-54487 


34 


.07987 


12.5199 


.09746 


10.2602 


.11511 


8.68701 


.13284 


7.52806 


35 


.08017 


12.4742 


.09776 


10.2294 


•11541 


8.66482 


.13313 


7-51132 


36 


.08046 


12.4288 


.09805 


10.1988 1 


.11570 


8.64275 


.13343 


7-49465 


37 


.08075 


12.3838 


.09834 


10.1683 


.11600 


8.62078 


.13372 


7.47806 


38 


.08104 


12.3390 


.09864 


IO.I381 , 


.11629 


8.59893 


.13402 


7-46154 


39 


.08134 


12.2946 


.09893 


10.1080 


•11659 


8.57718 


•13432 


7-44509 


40 


.08163 


12.2505 


.09923 


10.0780 


.11688 


8-55555 


.13461 


7.42871 


41 


.08192 


12.2067 


.09952 


10.0483 


.11718 


8.53402 


•13491 


7.41240 


42 


.08221 


12.1632 


.09981 


10.0187 


.11747 


8.51259 


•13521 


7-39616 


43 


.08251 


I2.I20I 


.10011 


9.98931 


.11777 


8.49128 


.13550 


7-37999 


44 


.08280 


12.0772 


.10040 


9.96007 


.11806 


8.47007 


1 .13580 


7-36389 


45 


.08309 


12.0346 


.10069 


9.93101 


.11836 


8.44896 


1 .13609 


7-34786 


46 


•08339 


11.9923 


.10099 


9.9021I 


.11865 


8.42795 


•13639 


7-33190 


47 


.08368 


11-9504 


.10128 


9-87338 


.11895 


8.40705 


.13669 


7.31600 


48 


.08397 


11.9087 


.10158 


9.84482 


.11924 


8.38625 


.13698 


7.30018 


49 


.08427 


11.8673 


.10187 


9.8164I 


.11954 


8.36555 


.13728 


7.28442 


5Q 


.08456 


11.8262 


.10216 


9.78817 


•11983 


8.34496 


.13758 


7-26873 


51 


.08485 


"•7853 


.10246 


9.76009 


.12013 


8.32446 


.13787 


7.25310 


52 


.08514 


11.7448 


.10275 


9.73217 


.12042 


8.30406 


.13817 


7-23754 


53 


.08544 


11.7045 


.10305 


9.70441 


.12072 


8.28376 


.13846 


7.22204 


54 


•08573 


11.6645 


•10334 


9.67680 


.12101 


8.26355 


.13876 


7.20661 


55 


.08602 


11.6248 


.10363 


9-64935 


.12131 


8.24345 


.13906 


7.19125 


56 


.08632 


11-5853 


•10393 


9.62205 


.12160 


8.22344 


•13935 


7^17594 


57 


.08661 


II.5461 


.10422 


9.59490 


.12190 


8.20352 


.13965 


7.16071 


58 


.08690 


11.5072 


.10452 


9.56791 


.12219 


8.18370 


•13995 


7-14553 


59 


.08720 


11.4685 


.10481 


9.54106 


.12249 


8.16398 


.14024 


7-13042 


60 


.08749 


1 1. 4301 


.10510 


9-51436 


.12278 


8.14435 


.14054 


7-11537 


/ 


Co-tan. 


Tan. 


,Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 




8 


5° 


1 


84? 


8 


3° 


8 


2° 



MEASUREMENT OF RIGHT TRIANGLES 



147 





8 





9 





10° 


IP 




/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. I Co-tan. 


f 


o 


.14054 


7-11537 


.15838 


6.31375 


•17633 


5-67128 


.19438 


5.14455 


6c 


I 


.14084 


7.10038 


.15868 


6.30189 


.17663 


5-66165 


.19468 


5-13658 


59 


2 


.14113 


7.08546 


.15898 


6.29007 


.17693 


5-65205 


.19498 


5.12862 


58 


3 


•14143 


7.07059 


.15928 


6.27829 


•17723 


5.64248 


.19529 


5.12069 


5-i 


4 


.14173 


7-05579 


.15958 


6.26655 


.17753 


5-63295 


.19559 


5-11279 


56 


5 


.14202 


7.04105 


.15988 


6.25486 


•17783 


5-62344 


.19589 


5.10490 


55 


6 


.14232 


7-02637 


.16017 


6.24321 


.17813 


5-61397 


.19619 


5.09704 


54 


7 


.14262 


7-01174 


.16047 


6.23160 


.17843 


5-60452 


.19649 


5.08921 


53 


8 


.14291 


6.99718 


.16077 


6.22003 


.17873 


5-59511 


.19680 


5. 08 1 39 


52 


9 


.14321 


6.98268 


.16107 


6.20851 


-17903, 


5-58573 


.19710 


5.07360 


51 


lO 


•14351 


6.96823 


• 16137 


6.19703 


•17933 


5-57638 


.19740 


5.06584 


5c 


n 


.14381 


6.95385 


.16167 


6.18559 


-17963 


5.56706 


.19770 


5. 05809 


49 


12 


.14410 


6.93952 


.16196 


6.17419 


•17993 


5.55777 


.19801 


5.05037 


48 


13 


.14440 


6.92525 


.16226 


6.16283 


.18023 


5.54851 


.19831 


5.04267 


47 


14 


.14470 


6.91 104 


.16256 


6.15151 


•18053 


5.53927 


.19861 


5.03499 


46 


IS 


.14499 


6.89688 


.16286 


6.14023 


.18083 


5-53007 


.19891 


5.02734 


45 


i6 


.14529 


6.88278 


.16316 


6.12899 


.18113 


5-52090 


.19921 


5-01971 


44 


17 


.14559 


6-86874 


,16346 


6.II779 


.18143 


5-51176 


.19952 


5.01210 


43 


i8 


.14588 


6.85475 


.16376 


6.10664 


.18173 


5-50264 


.19982 


5.00451 


42 


19 


.14618 


6.840S2 


.16405 


6.09552 


.18203 


5-49356 


.20012 


4.99695 


41 


20 


.14648 


6.82694 


.16435 


6.08444 


.18233 


5-48451 


.20042 


4.98940 


40 


21 


.14678 


6.81312 


.16465 


6.07340 


.18263 


5-47548 


.20073 


4.98188 


39 


22 


.14707 


6.79936 


.16495 


6.06240 


.18293 


5-46648 


.20103 


4-97438 


38 


23 


.14737 


6.78564 


.16525 


6.05143 


.18323 


5-45751 


.20133 


4.96690 


37 


24 


.14767 


6.77199 


.16555 


6.04051 


.18353 


5-44857 


.20164 


4-95945 


36 


25 


.14796 


6.75838 


.16585 


6.02962 


.18383 


5-43966 


.20194 


4-95201 


35 


26 


.14826 


6.74483 


.16615 


6.01878 


.18414 


5-43077 


.20224 


4.94460 


34 


27 


.14856 


6.73133 


.16645 


6.00797 


.18444 


5.42192 


.20254 


4-93721 


33 


28 


.14886 


6.71789 


.16674 


5.99720 


.18474 


5-41309 


.20285 


4.92984 


32 


29 


.14915 


6.70450 


.16704 


5.98646 


.18504 


5-40429 


.20315 


4.92249 


31 


30 


•14945 


6.69116 


•16734 


5-97576 


.18534 


5-39552 


.20345 


4.91516 


30 


31 


.14975 


6.67787 


.16764 


5-96510 


.18564 


5-38677 


•20376 


4.90785 


29 


32 


.15005 


6.66463 


.16794 


5-95448 


.18594 


5-37805 


.20406 


4.90056 


28 


33 


.15034 


6.65144 


.16824 


5-94390 


.18624 


5-36936 


.20436 


4.89330 


27 


34 


.15064 


6.63831 


.16854 


5-93335 


.18654 


5.36070 


.20466 


4-83605 


2d 


35 


.15094 


6.62523 


.16884 


5-92283 


.18684 


5-35206 


.20497 


4.87882 


25 


36 


.15124 


6.61219 


.16914 


5-91235 


.18714 


5-34345 


.20527 


4.87162 


24 


37 


.15153 


6.59921 


.16944 


5-90191 


.18745 


5.33487 


.20557 


4.86444 


23 


38 


.15183 


6.58627 


•16974 


5-89151 


.18775 


5-32631 


.20588 


4.85727 


22 


39 


.15213 


6.57339 


.17004 


5.88114 


.18805 


5-31778 


.20618 


4-85013 


21 


40 


.15243 


6.56055 


•17033 


5.87080 


.18835 


5.30928 


.20648 


4.84300 


20 


41 


.15272 


6.54777 


.17063 


5-86051 


.18865 


5.30080 


.20679 


483590 


ig 


42 


.15302 


6.53503 


• 17093 


5-85024 


.18895 


5.29235 


.20709 


4.82882 


18 


43 


.15332 


6.52234 


.17123 


5.84001 


.18925 


5.28393 


.20739 


4-82175 


17 


44 


.15362 


6.50970 


•17153 


5.82982 


.18955 


5.27553 


.20770 


4.81471 


16 


45 


.15391 


6.49710 


.17183 


5.81966 


.18986 


5.26715 


.20800 


4.80769 


15 


46 


.15421 


6.48456 


.17213 


5-80953 


.19016 


5-25880 


.20830 


4.80068 


14 


47 


.15451 


6.47206 


.17243 


5-79944 


.19046 


5.25048 


.20861 


4-79370 


13 


48 


.15481 


6.45961 


.17273 


5.78938 


.19076 


5.24218 


.20891 


4-78673 


12 


49 


.15511 


6.44720 


.17303 


5-77936 


.19106 


5-23391 


.20921 


4.77978 


II 


50 


.15540 


6.43484 


.17333 


5.76937 


.19136 


5.22566 


.20952 


4.77286 


10 


51 


.15570 


6.42253 


.17363 


5-75941 


.19166 


5-21744 


.20982 


4.76595 


9 


52 


.15600 


6.41026 


.17393 


5-74949 


.19197 


5.20925 


.21013 


4.75906 


8 


53 


.15630 


6.39804 


.17423 


5-73960 


.19227 


5.20107 


.21043 


4.75219 


7 


54 


.15660 


6.38587 


.17453 


5-72974 


.19257 


5.19293 


•21073 


4-74534 


6 


55 


.15689 


6.37374 


•17483 


5.71992 


.19287 


5.18480 


.21104 


4-73851- 


5 


56 


.15719 


6.36165 


.17513 


5-71013 


.19317 


5.17671 


.21134 


4.73170 


4 


57 


.15749 


6.34961 


.17543 


5-70037 


• 19347 


5-16863 


.21164 


4.72490 


3 


58 


.15779 


6.33761 


.17573 


>. 69064 


.19378 


5-16058 


.21195 


4-71813 


2 


59 


.15809 


6.32566 


.17603 


5-68094 


.19408 


5.15256 


.21225 


4-71137 


I 


60 


.15838 


6-31375 


•17633 


5.67128 


•19438 


5.14455 


.21256 


4-70463 





/ 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


/ 


8 


1° 


8( 


r 


7 


9° 


7< 


s° 





148 



MECHANICS AND ALLIED SUBJECTS 





12° 


13° 


14° 


15° f 


/ 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


o 


.21256 


4.70463 


.23087 


4.33148 


.24933 


4.01078 


.26795 


3.73205 


I 


.21286 


4.69791 


'.23117 


4.32573 


.24964 


4.00582 


.26826 


3.72771 


2 


.21316 


4.69121 


.23148 


4^32001 


.24995 


4.0C086 


.26857 


3-72338 


3 


.21347 


4.68452 


.23179 


4.31430 


.25026 


399592 


.26888 


3.71907 


4 


.21377 


4.67786 


.23209 


4.30860 


.25056 


3.99099 


.26920 


3.71476 


5 


.21408 


4.67121 


•23240 


4-30291 


.25087 


3.98607 


•26951 


3.71046 


6 


.21438 


4.66458 


.23271 


4.29724 


.25118 


3.98117 


.26982 


3.70616 


7 


.21469 


4-65797 


.23301 


4.29159 


.25149 


3.97627 


•27013 


3.70188 


8 


.21499 


4.65138 


'23332 


4.28595 


.25180 


397139 


.27044 


3-69761 


9 


.21529 


4.64480 


■233(>3 


4.28032 


.25211 


3-96651 


.27076 


3-69335 


lO 


.21560 


4.63825 


■23393 


4.27471 


.25242 


3.96165 


.27107 


3-68909 


II 


.21590 


4-63171 


.23424 


4.26911 


.25273 


3.95680 


.27138 


3-68485 


12 


.21621 


4.62518 


.23455 


4.26352 


•25304 


3-95196 


.27169 


3-68061 


13 


.21651 


4.61868 


.23485 


4^25795 


•25335 


3-94713 


.27201 


3-67638 


14 


.21682 


4.61219 


•23516 


4.25239 


.25366 


3-94232 


.27232 


3-67217 


15 


.21712 


4-60572 


.23547 


4.24685 


.25397 


3.93751 


.27263 


3-66796 


i6 


.21743 


4-59927 


•23578 


4-24132 


.25428 


3.93271 


.27294 


3.66376 


17 


.21773 


4-59283 


.23608 


4-23580 


.25459 


3-92793 


.27326 


3-65957 


i8 


.21804 


4.58641 


.23639 


4.23030 


.25490 


3.92316 


.27357 


3-65538 


19 


.21834 


4.58001 


.23670 


4.22481 


.25521 


3-91839 


.27388 


3-65121 


20 


.21864 


4-57363 


.23700 


4.21933 


.25552 


3.91364 


.27419 


3.64705 


21 


.21895 


4-56726 


.23731 


4.21387 


.25583 


3.90890 


•27451 


3.64289 


22 


.21925 


4.56091 


.23762 


4.20842 


.25614 


3.90417 


.27482 


3.63874 


23 


.21956 


4-55458 


.23793 


4.20298 


.25645 


389945 


.27513 


3.63461 


24 


.21986 


4.54826 


-23823 


4.19756 


.25676 


3-89474 


.27545 


3.63048 


25 


.22017 


4.54196 


•23854 


4.19215 


.25707 


3.89004 


.27576 


3.62636 


26 


.22047 


4.53568 


.23885 


4.18675 


.25738 


3.88536 


.27607 


3.62224 


27 


.22078 


4.52941 


.23916 


4.18137 


.25769 


3.88068 


.27638 


3.61814 


28 


.22108 


4.52316 


.23946 


4.17600 


.25800 


3.87601 


.27670 


3.61405 


29 


.22139 


4-51693 


.23977 


4.17064 


.25831 


3.87136 


.27701 


3-60996 


30 


.22169 


4.51071 


.24008 


4.16530 


.25862 


3.86671 


.27732 


3-60588 


31 


.22200 


4-50451 


.24039 


4.15997 


.25893 


3.86208 


.27764 


3.60181 


32 


.22231 


4.49832 


.24069 


4.15465 


.25924 


3-85745 


.27795 


3-59775 


33 


.22261 


4.49215 


.24100 


4.14934 


.25955 


3-85284 


.27826 


3-59370 


34 


.22292 


4.48600 


.24131 


4.14405 


.25986 


3.84824 


.27858 


3.58966 


35 


.22322 


4.47986 


.24162 


4.13877 


.26017 


3.84364 


.27889 


3-58562 


36 


.22353 


4-47374 


.24193 


4-13350 


.26048 


3.83906 


.27920 


3.58160 


37 


.22383 


4*46764 


.24223 


4.12825 


.26079 


3-83449 


..27952 


3-57758 


38 


.22414 


4-46155 


.24254 


4.12301 


.26110 


3.82992 


.27983 


3-57357 


39 


.22444 


4-45548 ] 


•24285 


4.11778 


.26141 


3-82537 


.28015 


356957 


40 


.22475 


4.44942 


.24316 


4.11256 


.26172 


3.82083 


.28046 


3-56557 


41 


.22505 


4-44338 


•24347 


4.10736 


.26203 


3-81630 


.28077 


3-56159 


42 


.22536 


4.43735 


•24377 


4.10216 


•26235 


3-81177 


.2S109 


3-55761 


43 


.22567 


4-43134 


.24408 


4.09699 


.26266 


3.80726 


.28140 


3-55364 


44 


.22597 


4-42534 


.24439 


4.09182 


.26297 1 


3.80276 


.28172 


3-54968 


45 


.22628 


4.41936 


•24470 


4.08666 


.26328 


3.79827 


.28203 


3-54573 


46 


.22658 


4.41340 


•24501 


4.08152 


.26359 


3-79378 


.28234 


3-54179 


47 


.22689 


4-40745 


'24532 


4-07639 


.26390 


3-78931 


.28266 


3-53785 


48 


.22719 


4.40152 


.24562 


4.07127 


.26421 1 


3-78485 


.28297 


3-53393 


49 


.22750 


4-39560 


.24593 


4.06616 


.26452 


3.78040 


.28329 


3-53001 


50 


.22781 


4.38969 


.24624 


4.06107 


.26483 


3-77595 


.28360 


3.52609 


51 


.22811 


4-38381 


•24655 


4-05599 


•26515 


3-77152 


•28391 


3.52219 


52 


.22842 


4-37793 


.24686 


4.05092 


.26546 


3.76709 


.28423 


3.51829 


53 


.22872 


4-37207 


.24717 


4.04586 


.26577 


3.76268 


•28454 


3.51441 


54 


.22903 


4.36623 


•24747 


4.04081 


.26608 


3.75828 


.28486 


3-51053 


55 


-22934 


4.36040 


.24778 


4-03578 


,26639 


3-75388 


.28517 


3.50666 


56 


.22964 


4-35459 


.24809 


4-03075 


.26670 


3.74950 


.28549 


3-50279 


57 


.22995 


4-34879 


.24840 


4-02574 


.26701 


3.74512 


.28580 


3.49894 


58 


.23026 


4-34300 


.24871 


4.02074 


.26733 


3.74075 


.28612 


3-49509 


59 


.23056 


4-33723 


.24902 


4.01576 


.26764 


3-73640 


.28643 


3-49125 


60 


.23087 


4-33148 


.24933 


4.01078 


.26795 


3.73205 


.28675 


3.48741 


/ 


Co-tan. 


Tan. 


Co-tan.' 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


1 


77 





7G 


° 


75 


►° 


74 


1° 



MEASUREMENT OF RIGHT TRIANGLES 



149 





1G° 


1 


7^^ 


18° 


19° 1 


/ 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


o 


.28675 


3-48741 


.30573 


3.27085 


.32492 


3-07768 


.34433 


2.90421 


I 


.28706 


3-48359 


.30605 


3-26745 


•32524 


3-07464 


.34465 


2.90147 


2 


.28738 


3-47977 


.30637 


3.26406 


.32556 


3.07160 


•34498 


2.89873 


3 


.28769 


3-47596 


.30669 


3.26067 


.32588 


3-06857 


.34530 


2.89600 


4 


.28800 


3-47216 


.30700 


3.25729 


.32621 


3-06554 


•34563 


2.89327 


5 


.28832 


3.46837 


.30732 


3-25392 


.32653 


3-06252 


•34596 


2.8905s 


6 


.28864 


3-46458 


.30764 


3-25055 


.32685 


3-05950 


•34628 


2.88783 


7 


.28895 


3.46080 


.30796 


3.24719 


.32717 


3-05649 


.34661 


2.885 1 1 


8 


.28927 


3-45703 


.30828 


3-24383 


.32749 


3-05349 


.34693 


2.88240 


9 


.28958 


3-45327 


.30860 


3.24049 


.32782 


3-05049 


.34726 


2.87970 


lO 


.28990 


3-44951 


.30891 


3-23714 


•32814 


3 -04749 


.34758 


2.87700 


II 


.29021 


3.44576 


.30923 


3-23381 


.32846 


3-04450 


.34791 


2.87430 


r2 


•29053 


3.44202 


.30955 


3.23048 


.32878 


3-04152 


.34824 


2.87161 


13 


.29084 


3-43829 


.30987 


3.22715 


.32911 


3-03854 


.34856 


2.86892 


14 


.29116 


3-43456 


.31019 


3-22384 


.32943 


3-03556 


.34889 


2.86624 


tS 


.29147 


3-43084 


.31051 


3-22053 


.32975 


3.03260 


.34922 


2.86356 


i6 


.29179 


3-42713 


.31083 


3.21722 


.33007 


3-02963 


.34954 


2.86089 


17 


.29210 


3-42343 


.31115 


3-21392 


.33040 


3.02667 


.34987 


2.85822 


i8 


.29242 


3-41973 


.31147 


3.21063 


-33072 


3-02372 


•35019 


2-85555 


19 


.29274 


3.41604 


.31178 


3-20734 


.33104 


3-02077 


.35052 


2.85289 


CO 


.29305 


3-41236 


.31210 


3.20406 


•33136 


3-01783 


.35085 


2.85023 


21 


.29337 


3-40869 


.31242 


3.20079 


.33160 


3-01489 


.35117 


2.84758 


22 


.29368 


3.40502 


•31274 


3^19752 


.33201 


3.01196 


.35150 


2.84494 


-"3 


.29400 


3.40136 


.31306 


3.19426 


-33233 


3-00903 


.35183 


2.84229 


24 


.29432 


3-39771 


•31338 


3.19100 


•33266 


3.0061 1 


.■35216 


2.83965 


25 


.29463 


3-39406 


•31370 


3-18775 


.33298 


3.00319 


.35248 


2.83702 


26 


•29495 


3-39042 


.31402 


3-18451 


'33330 


3.00028 


.35281 


2-83439 


27 


.29526 


3-38679 


.31434 


3-18127 


-333^3 


2.99738 


.35314 


2.83176 


28 


.29558 


3-38317 


.31466 


3^17804 


.33395 


2-99447 


.35346 


2.82914 


29 


.29590 


3-37955 


.31498 


3-17481 


.33427 


2.99158 


-35379 


2.82653 


30 


.29621 


3-37594 


.31530 


3-17159 


•33460 


2.98868 


•35412 


2.82391 


31 


.29653 


3-37234 


.31562 


3-16838 


.33492 


2.98580 


.35445 


2.82130 


32 


.29685 


3-36875 


.31594 


3-16517 


•33524 


2.98292 


.35477 


2.81870 


33 


.29716 


3-36516 


.31626 


3.16197 


.33557 


2.98004 


.35510 


2.81610 


34 


.29748 


3-36158 


•31658 


3-15877 


.33589 


2.97717 


.35543 


2.81350 


35 


.29780 


3-35800 


•31690 


3-15558 


.33621 


2-97430 


•35576 


2.81091 


36 


.29811 


3-35443 


.31722 


3-15240 


.33654 


2.97144 


•35608 


2.80833 


37 


.29843 


3-35087 


.31754 


3-14922 


.33686 


2.96858 


.35641 


2.80574 


38 


.29875 


3-34732 


.31786 


3.14605 


.33718 


2.96573 


.35674 


2.80316 


39 


.29906 


3-34377 


.31818 


3.14288 


•33751 


2.96288 


.35707 


2.80059 


40 


.29938 


3-34023 


.31850 


3.13972 


.33783 


2.96004 


.35740 


2.79802 


41 


.29970 


3-33670 


.31882 


3.13656 


.33816 


2.95721 


.35772 


2.79545 


42 


.30001 


3-33317 


•319T4 


3-13341 


.33848 


2-95437 


.35805 


2.79289 


43 


.30033 


3-32965 


.31946 


5.13027 


.33881 


2.95155 


.35838 


2.79033 


44 


.30065 


3-32614 


•31978 


3-12713 


•33913 


2.94872 


.35871 


2.78778 


45 


.30097 


3.32264 


.32010 


3.12400 


•33945 


2.94590 


.35904 


2-78523 


46 


.30128 


3-31914 


.32042 


3.12087 


.33978 


2.94309 


.35937 


2.78269 


47 


.30160 


3-31565 


•32074 


3-11775 


.34010 


2.94028 


.35969 


2.78014 


48 


.30192 


3-31216 


.32106 


3.11464 


.34043 


2.93748 


.36002 


2.77761 


49 


.30224 


3-30868 


•32139 


3-11153 


•34075 


2.93468 


•3603 s 


2.77507 


50 


•30255 


3-30521 


.32171 


3.10842 


.34108 


2.93189 


.36068 


2^77254 


51 


.30287 


3-30174 


.32203 


3-10532 


.34140 


2.92910 


.36101 


2.77002 


52 


•30319 


3.29829 


•32235 


3-I0223 


.34173 


2.92632 


•36134 


2.76750 


53 


•30351 


3-29483 


.32267 


3.09914 


.34205 


2.92354 


•36167 


2.76498 


54 


.30382 


3-29139 


.32299 


3.09606 


.34238 


2.92076 


•36199 


2.76247 


55 


.30414 


3-28795 


-32331 


3.09298 


.34270 


2.91799 


•36232 


£.75996 


56 


.30446 


3.28452 


-32363 


3.08991 


.34303 


2.91523 


•36265 


2.75746 


57 


.30478 


3.28109 


.32396 


3-08685 


.34335 


2.91246 


.36298 


2.75496 


58 


•30509 


3-27767 


.32428 


3-08379 


.34368 


2.90971 


•36331 


2.75246 


59 


.30541 


3.27426 


.32460 


3.08073 


.34400 


2.90696 


•36364 


2.74997 


60 


.30573 


3.27085 


.32492 


3-07768 


-34433 


2.90421 


•36397 


2.74748 


/ 


Co-tax. 


Tan. 


Co-tax. 1 Tax. 


Co-tan. 


Tax. 


Co-tan. 


Tan. 




7 


3° 


7 


2© 


7 


P 


1 7 


0° 



60 

59 
58 
57 

56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 

21 
20 

19 
18 

17 
16 
15 
14 
13 
12 
II 
10 

9 

8 

7 
6 

5 
4 
3 

2 
I 
c 



150 



MECHANICS AND ALLIED SUBJECTS 





20° 


2P 


22° 


23° 




/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


/ 


o 


.36397 


2.74748 


•38386 


2.60509 


•40403 


2.47509 


.42447 


2.35585 


6a 


I 


•36430 


2.74499 


•38420 


2.60283 


.40436 


2.47302 


.42482 


2.35395 


55 


2 


•36463 


2.74251 


•38453 


2.60057 


•40470 


2.47095 


•42516 


2.35205 


58 


3 


.36496 


2.74004 


.38487 


2.59831 


.40504 


2.46888 


.42551 


2.35015 


57 


4 


.36529 


2.73756 


.38520 


2.59606 


•40538 


2.46682 


.42585 


2.34825 


56 


5 


.36562 


2.73509 


.38553 


2.59381 


•40572 


2.46476 


.42619 


2.34636 


55 


6 


•36595 


2.73263 


.38587 


2.59156 


.40606 


2.46270 


.42654 


2.34447 


54 


7 


.36628 


2.73017 


.38620 


2.58932 


.40640 


2.46065 


.42688 


2.34258 


53 


8 


.36661 


2.72771 


•38654 


2.58708 


.40674 


2.45860 


.42722 


2.34069 


52 


9 


.36694 


2.72526 


.38687 


2.58484 


•40707 


2.45655 


•42757 


2.33881 


51 


lO 


.36727 


2.72281 


.38721 


2.58261 


.40741 


2.45451 


.42791 


2.33693 


5c 


II 


.36760 


2.72036 


•38754 


2.58038 


•40775 


2.45246 


.42826 


2.3350s 


49 


12 


•36793 


2.71792 


•38787 


2.57815 


.40809 


2.45043 


.42860 


2.33317 


48 


13 


.36826 


2.71548 


.38S21 


2.57593 


.40843 


2.44839 


.42894 


2.33130 


47 


14 


.36859 


2.71305 


.38854 


2.57371 


.40877 


2.44636 


.42929 


2.32943 


46 


15 


.36892 


2.71062 


.38888 


2.57150 


.40911 


2.44433 


.42963 


2.32756 


45 


16 


.36925 


2.70819 


.38921 


2.56928 


•40945 


2.44230 


.42998 


2.32570 


44 


17 


.36958 


2.70577 


.38955 


2.56707 


•40979 


2.44027 


.43032 


2.32383 


43 


18 


.36991 


2.70335 


.38988 


2.56487 


.41013 


2.4382s 


.43067 


2.32197 


42 


19 


.37024 


2.70094 


.39022 


2.56266 


.41047 


2.43623 


.43101 


2.32012 


41 


20 


.37057 


2.69853 


•39055 


2.56046 


.41081 


2.43422 


.43136 


2.31826 


40 


21 


.37090 


2.69612 


.39089 


2.55827 


.41115 


2.43220 


.43170 


2.3164I 


39 


22 


.37124 


2.69371 


.39122 


2.55608 


.41149 


2.43019 


.43205 


2.31456 


38 


23 


.37157 


2.6913I 1 


.39156 


2.55389 


.41183 


2.42819 


.43239 


2.31271 


31 


24 


.37190 


2.68892 : 


.39190 


2.55170 


.41217 


2.42618 


.43274 


2.31086 


36 


25 


.37223 


2.68653 


.39223 


2.54952 


.41251 


2.4241S 


.43308 


2.30902 


35 


26 


.37256 


2.68414 


.39257 


2.54734 


.41285 


2.42218 


.43343 


2.30718 


34 


27 


.37289 


2.:8i7s ; 


•39290 


2.54516 


.41319 


2.42019 


.43378 


2.30534 


33 


28 


•37322 


2.67937 i 


•39324 


2.54299 


.41353 


2.41819 


.43412 


2.30351 


32 


29 


.37355 


2.67700 j 


•39357 


2.54082 


.41387 


2.41620 


.43447 


2.30167 


31 


30 


.37388 


2.67462 


.39391 


2.53865 


.41421 


2.41421 


.43481 


2.29984 


30 


31 


.37422 


2.67225 


.39425 


2.53648 


.41455 


2.41223 


.43516 


2.29801 


29 


32 


•37455 


2.66989 


•39458 


2.53432 


.41490 


2.41025 


.43550 


2.29619 


28 


33 


.37488 


2.66752 


•39492 


2.53217 


.41524 


2.40827 


.43585 


2.29437 


27 


34 


.37521 


2.66516 


•39526 


2.53001 


.41558 


2.40629 


.43620 


2.29254 


26 


35 


.37554 


2.66281 


•39559 


2.52786 


.41592 


2.40432 


.43654 


2.29073 


25 


36 


.37588 


2.66046 


•39593 


2.52571 


.41626 


2.40235 


.43689 


2.28891 


24 


37 


.37621 


2.658II 


.39626 


2.52357 


■ .41660 


2.40038 


.43724 


2.28710 


23 


38 


.37654 


2.65576 


.39660 


2.52142 


.41694 


2.39841 


.43758 


2.28528 


22 


39 


.37687 


2.65342 


.39694 


2.51929 


.41728 


2.39645 


.43793 


2.28348 


21 


40 


.37720 


2.65109 


.39727 


2.51715 


.41763 


2.39449 


.43828 


2.28167 


20 


41 


.37754 


2.64875 


.39761 


2.51502 


•41797 


2.39253 


.43862 


2.27987 


19 


42 


.37787 


2.64642 


•39795 


2.51289 


.41831 


2.39058 


.43897 


2.27806 


18 


43 


•37820 


2.64410 


•39829 


2.51076 


.41865 


2.38862 


.43932 


2.27626 


17 


44 


•37853 


2.64177 


.39862 


2.50864 


.41899 


2.38668 


.43966 


2.27447 


16 


45 


•37387 


2.63945 


.39896 


2.50652 


•41933 


2.38473 


.44001 


2.27267 


15 


46 


•37920 


2.63714 


•39930 


2.50440 


.41968 


2.38279 


.44036 


2.27088 


14 


47 


•37953 


2.63483 


.39963 


2.50229 


.42002 


2.38084 


.44071 


2.26909 


13 


48 


•37986 


2.63252 


•39997 


2.50018 


.42036 


2.37891 


.44105 


2.26730 


12 


49 


.3S020 


2.63021 


.40031 


2.49807 


.42070 


2.37697 


.44140 


2.26552 


II 


50 


.38053 


2.62791 


.40065 


2.49597 


•42105 


2.37504 


.44175 


2.26374 


10 


51 


.38086 


2.62561 


.40098 


2.49386 


•42139 


2.37311 


.44210 


2.26196 


9 


52 


.38120 


2.62332 


.40132 


2.49177 


.42173 


2.37118 


•44244 


2.26018 


8 


53 


•38153 


2.62103 


.40166 


2.48967 


.42207 


2.36925 


.44279 


2.25840 


7 


54 


.38186 


2.61874 


.40200 


2.48758 


.42242 


2.36733 


•44314 


2.25663 


6 


55 


.38220 


2.61646 


•40234 


2.48549 


.42276 


2.36541 


•44349 


2.25486 


5 


56 


.38253 


2.6I4I8 


.40267 


2.48340 


.42310 


2.36349 


•44384 


2.25309 


4 


57 


.38286 


2.6II90 


.40301 


2.48132 


•42345 


2.36158 


.44418 


2.25132 


3 


58 


.38320 


2.60963 


•40335 


2.47924 


•42379 


2.35967 


•44453 


2.24956 


2 


59 


•38353 


2.60736 


•40369 


2.47716 


.42413 


2.35776 


.44488 


2.24780 


I 


60 


.38386 


2.60509 


.40403 


2.47509 


•42447 


2.35585 


.44523 


2.24604 





f 


Co-TAN. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


/ 




61 


r 


6^ 


50 


6' 


r° 


61 


5° 





MEASUREMENT OF RIGHT TRIANGLES 



151 





24° 1 


25^ 1 


26° 


27° 




/ 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 




o 


•44523 


2.24604 


.46631 


2.14451 


•48773 


2.05030 


.50953 


1. 9626 1 


60 


I 


.44558 


2.24428 


.46666 


2.14288 


.48809 


2.04879 


.50989 


1. 96 1 20 


59 


2 


•44593 


2.24252 


.46702 


2.14125 


•48845 


2.04728 


.51026 


1-95979 


53 


3 


.44627 


2.24077 


.46737 


2.13963 


.48881 


2.04577 


.51063 


1.95838 


57 


4 


.44662 


2.23902 


.46772 


2.13801 


.48917 


2.04426 


•51099 


1.95698 


56 


5 


•44697 


2.23727 


.46808 


2.13639 


•48953 


2.04276 


.51136 


1-95557 


55 


6 


•44732 


2.23553 


.46843 


2.13477 


.48989 


2.04125 


.51173 


1-95417 


54 


7 


.44767 


2.23378 


.46879 


2.13316 


.49026 


2.03975 


.51209 


1.95^*77 


53 


8 


.44802 


2.23204 


.46914 


2^i3i54 


.49062 


2.03825 


.51246 


1-95137 


52 


9 


.44837 


2.23030 


.46950 


2.12993 


.49098 


2.03675 


.51283 


1.94997 


51 


lO 


.44872 


2.22857 


.46985 


2.12832 


•49134 


2.03526 


.51319 


1.94858 


50 


II 


.44907 


2.22683 


.47021 


2.12671 


.49170 


2.03376 


.51356 


1.94718 


49 


12 


.44942 


2.22510 


.47056 


2.12511 


.49206 


2.03227 


.51393 


1-94579 


48 


13 


•44977 


2.22337 


.47092 


2.12350 


.49242 


2.03078 


.51430 


1.94440 


47 


14 


.45012 


2.22164 


.47128 


2.12190 


.49278 


2.02929 


.51467 


1. 9430 1 


46 


15 


.45047 


2.21992 


•47163 


2.12030 


•49315 


2.02780 


.51503 


1.94162 


45 


i6 


.45082 


2.21819 


.47199 


2.11871 


•49351 


2.02631 


.51540 


1.94023 


44 


17 


•45117 


2.21647 


•47234 


2.11711 


•49387 


2.02483 


.51577 


1.93885 


43 


i8 


•45152 


2.21475 


.47270 


2.11552 


•49423 


2.02335 


•51614 


1-93746 


42 


19 


.45187 


2.21304 


•47305 


2.11392 


•49459 


2.02187 


•51651 


1.93608 


41 


20 


.45222 


2.21132 


.47341 


2.11233 


•49495 


2.02039 


.51688 


1.93470 


40 


21 


.45257 


2.20961 


•47577 


2.11075 


•49532 


2.01891 


•51724 


1.93332 


39 


22 


.45292 


2.20790 


.47412 


2.10916 


•49568 


2.01743 


.51761 


1-93195 


38 


23 


•45327 


2.20619 


•47448 


2.10758 


.49604 


2.01596 


•51798 


1-93057 


37 


24 


.45362 


2.20449 


.47483 


2.10600 


.49640 


2.01449 


.51835 


1.92920 


36 


25 


.45397 


2.20278 


•47519 


2.10442 


.49677 


2.01302 


.51872 


1.92782 


35 


26 


.45432 


2.20108 


.47555 


2.10284 


•49713 


2.OI155 


.51909 


1.92645 


34 


27 


.45467 


2.19938 


.47590 


2.10126 


.49749 


2.01008 


.51946 


1.92508 


33 


28 


•45502 


2.19769 


.47626 


2.09969 


.49786 


2.00862 


.51983 


1-92371 


32 


29 


•45537 


2.19599 


.47662 


2. 098 II 


.49822 


2.00715 


.52020 


1.92235 


31 


30 


.45573 


2.19430 


.47698 


2.09654 


•49858 


2.00569 


.52057 


1.92098 


30 


31 


.45608 


2.19261 


.47733 


2.09498 


•49894 


2.00423 


•52094 


1. 91962 


29 


32 


.45643 


2.19092 


•47769 


2.09341 


•49931 


2.00277 


•52131 


1.91826 


28 


33 


.45678 


2.18923 


.47805 


2.09184 


.49967 


2.OO131 


.52168 


1. 9 1 690 


27 


34 


.45713 


2.18755 


.47840 


2.09028 


.50004 


1.99986 


.52205 


I-91554 


26 


35 


.45748 


2.18587 


•47876 


2.08872 


.50040 


1. 99841 


.52242 


1.91418 


25 


36 


.45784 


2.18419 


.47912 


2.08716 


.50076 


1.99695 


.52279 


1.91282 


24 


37 


.45819 


2.18251 


•47948 


2.08560 


•50113 


1.99550 


.52316 


1.91147 


23 


38 


.45854 


2.18084 


.47984 


2.08405 


•50149 


1.99406 


•52353 


1.91012 


22 


39 


.45S8Q 


2.17916 


.48019 


2.08250 


•50185 


1. 99261 


•52390 


1.90876 


21 


40 


.45924 


2.17749 


•48055 


2.08094 


.50222 


I.99116 


•52427 


1.90741 


20 


41 


.45960 


2.17582 


.48091 


2.07939 


.50258 


1.98972 


.52464 


1.90607 


19 


42 


•45995 


2.17416 


.48127 


2.07785 


•50295 


I.9S828 


.52501 


1.90472 


18 


43 


.46030 


2.17249 


.48163 


2.07630 


•50331 


1.98684 


•52538 


1.90337 


17 


44 


.46065 


2.17083 


.48198 


2.07476 


•50368 


1.98540 


.52575 


1.90203 


16 


45 


.46101 


2.16917 


.48234 


2.07321 


.50404 


1.98396 


.52613 


1 .90069 


15 


46 


.46136 


2.16751 


.48270 


2.07167 


.50441 


1.98253 


.52650 


1-89935 


14 


47 


.46171 


2.16585 


.48306 


2.07014 


.50477 


1. 981 10 


.52687 


1. 8980 1 


13 


48 


.46206 


2.16420 


.48342 


2.06860 


•50514 


1.97966 


.52724 


1.89667 


12 


<9 


.46242 


2.16255 


.48378 


2.06706 


•50550 


1.97823 


.52761 


1-89533 


II 


50 


•46277 


2.16090 


.48414 


2.06553 


.50587 


1.97680 


.52798 


1.89400 


10 


51 


•46312 


2.15925 


.48450 


2.06400 


.50623 


1.97538 


.52836 


1.89266 


9 


52 


•46348 


2.15760 


.48486 


2.06247 


.50660 


1.97395 : 


.52873 


I-89133 


8 


53 


•46383 


2.15596 


.48521 


2.06094 


.50696 


1.97253 ; 


.52910 


1 .89000 


7 


54 


.46418 


2.15432 


.48557 


2.05942 


.50733 


1. 971 1 1 


.52947 


1.88867 


6 


55 


•46454 


2.15268 


.48593 


2.05790 


.50769 


1.96969 


.52984 


1-88734 


5 


56 


.46489 


2.15104 


.48629 


2.05637 


.50806 


1.96827 


.53022 


1.88602 


4 


57 


.46525 


2.14940 


.48665 


2.05485 


.50843 


1.96685 


.53059 


1.88469 


3 


58 


.46560 


2.14777 


.48701 


2.05333 


.50879 


1.96544 


•53096 


1-88337 


2 


59 


.46595 


2.14614 


.48737 


2.05182 


.50916 


1 .96402 


.53134 


1.88205 


I 


60 


.46631 


2.14451 


.48773 


2.05030 


•50953 


1. 96261 


.53171 


1.88073 





/ 


Co-tan . 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-tan. 


Tan. 


/ 




6 


5° 


6 


40 


1 6 


3° 


6 


2° 





152 



MECHANICS AND ALLIED SUBJECTS 





28° 1 


29° I 


30° 


31° 






Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 




o 


•53171 


1.88073 


•55431 


1.80405 


•57735 


1-73205 


.60086 


1.66428 


6q 


I 


.53208 


1.87941 


•55469 


1. 80281 


•57774 


1.73089 


.60126 


1.663x8 


59 


2 


•53246 


1.87809 


•55507 


1. 80 1 58 


•57813 


1-7-973 


.60165 


I. 6209 


58 


3 


.53283 


1.87677 


•55545 


1.80034 


•57851 


1.72857 


.60205 


1.66099 


57 


4 


.53320 


1.87546 


•55583 


1.79911 


•57890 


1.72741 


.60245 


1-65990 


56 


5 


.53358 


1.87415 


.55621 


1.79788 


•57929 


1.72625 


.60204 


1.65881 


55 


6 


.53395 


1.87283 


•55659 


1.79665 


•57968 


1.72509 


.60324 


1.65772 


54 


7 


•53432 


1.87152 


.55697 


1.79542 


•58007 


1-72393 


.60364 


1.65663 


53 


8 


•53470 


1. 8702 1 


.55736 


I. 79419 


.58046 


1.72278 


.60403 


1.65534 


52 


9 


.53507 


1. 86891 


•55774 


1.79296 


•58085 


1.72163 


.60443 


1.65445 


51 


lO 


•53545 


1.86760 


•55812 


1.79174 


.58124 


1.72047 


.604S3 


1.65337 


50 


II 


.53582 


1.86630 


•55850 


1. 7905 1 


.58162 


1.71932 


.60522 


1.65228 


49 


12 


.53620 


1.86499 


.55888 


1.78929 


.58201 


1.71817 


.60562 


1. 65 1 20 


48 


13 


•53657 


1.86369 


.55926 


1.78807 


.58240 


1.71702 


.60602 


1.65011 


47 


-4 


.53694 


1.86239 


•55964 


1.78685 


.58279 


1.71588 


.60642 


1.64903 


46 


^5 


.53732 


1.86109 


.56003 


1^78563 


.58318 


1-71473 


.60681 


1.64795 


45 


i6 


.53769 


1.85979 


.56041 


1. 7844 1 


.58357 


1-71358 


.60721 


1.64687 


44 


17 


.53807 


1.85850 


.56079 


1.78319 


.58396 


1.71244 


.60761 


1.64579 


43 


i8 


.53844 


1.85720 


.56117 


1.78198 


.58435 


1.71129 


.60801 


1.64471 


42 


19 


.53882 


1.85591 


.56156 


1.78077 


.58474 


1.71015 


.60841 


1-64363 


41 


20 


.53920 


1.85462 


.56194 


1^77955 


.58513 


1. 70901 


.60881 


1.64256 


40 


21 


.53957 


1-85333 


.56232 


i^77834 


.58552 


1.70787 


.60921 


1. 64 1 48 


39 


22 


.53995 


1.85204 


.56270 


1-77713 


.58591 


1.70673 


.60960 


1. 64041 


38 


23 


.54032 


1.S5075 


.56309 


1.77592 


.58631 


1.70560 


.61000 


1-63934 


37 


24 


.54070 


1.84946 


-56347 


1.77471 


.58670 


1.70446 


.61040 


1.63826 


36 


25 


.54107 


1. 848 1 8 


.56385 


1-77351 


.58709 


1.70332 


.61080 


1.63719 


35 


26 


.54145 


1.84689 


-56424 


1.77230 


.58748 


1. 70219 


.61120 


1.63612 


34 


27 


■54183 


1.84561 


.56462 


1.77110 


.58787 


1.70106 


.61160 


1-63505 


33 


28 


.54220 


1-84433 


-56500 


1.76990 


.58826 


1.69992 


.61200 


1.63398 


32 


29 


•54258 


1-84305 


-56539 


1.76869 


.58865 


1.69879 


.61240 


1.63292 


31 


30 


.54296 


1.84177 


.56577 


1.76749 


.58904 


1.69766 


.61280 


1. 63 185 


30 


31 


•54333 


1.84049 


.56616 


1.76630 


.58944 


1-69653 


.61320 


1.63079 


29 


32 


.54371 


1.83922 


.56654 


1. 76510 


.58983 


I -69541 


.61360 


1.62972 


28 


33 


•54409 


1.83794 


.56693 


1.76390 


.59022 


1.69428 


.61400 


1.62866 


27 


34 


•54446 


1.83667 


.56731 


1.76271 


.59061 


1. 693 1 6 


.61440 


1.62760 


26 


35 


•54484 


1.83540 


.56760 


1.76151 


.59101 


1.69203 


.61480 


1.62654 


25 


36 


•54522 


1-83413 


.56808 


1-76032 


.59140 


1. 6909 1 


.61520 


1.62548 


24 


37 


•54560 


1.83286 


.56846 


1.75913 


.59179 


1.68979 


.61561 


1.62442 


23 


38 


•54597 


1-83159 


.56885 


1.75794 


.59218 


1.68866 


.61601 


1.62336 


22 


39 


•54635 


1-83033 


.56923 


1.75675 


.59258 


1.68754 


.61641 


1.62230 


21 


40 


•54673 


1.82906 


.56962 


1.75556 


-59297 


1.68643 


.61681 


1.62125 


20 


41 


•54711 


1.82780 


.57000 


1.75437 


-59336 


1. 6853 1 


.61721 


1.62019 


19 


42 


.54748 


1.82654 


.57039 


1.75319 


•59376 


1. 684 1 9 


.61761 


1.61914 


18 


43 


.54786 


1.82528 


.57078 


1.75200 


•59415 


1.68308 


.61801 


1.61808 


17 


44 


.54824 


1.82402 


.57116 


1.75082 


.59454 


1. 68 1 96 


.61842 


1. 6 1 703 


16 


45 


.54862 


1.82276 


.57155 


1.74964 


•59494 


1.68085 


.61882 


1.61598 


15 


46 


.54900 


1.82150 


-57193 


1.74846 


•59533 


1.67974 


.61922 


1. 6 1 493 


14 


47 


.54938 


1.82025 


-57232 


1.74728 


•59573 


1.67863 


.61962 


1.61388 


13 


48 


.54975 


1.81899 


-57271 


1.74610 


.59612 


1.67752 


.62003 


1.61283 


12 


49 


.55013 


1.81774 


.57309 


1.74492 


.59651 


1. 67641 


.62043 


1. 61 1 79 


II 


50 


•55051 


1. 8 1 649 


.57348 


1-74375 


.59691 


1-67530 


.62083 


1.61074 


10 


51 


.55089 


1.81524 


•57386 


1.74257 


-59730 


1. 67419 


.62124 


1.60970 


9 


52 


•55127 


1. 8 1 399 


-57425 


1.74140 


-59770 


1.67309 


.62164 


1.6086s 


8 


S3 


.55165 


1.81274 


•57464 


1.74022 


.59809 


1.67198 


.62204 


1.60761 


7 


54 


.55203 


1. 81 1 50 


.5-/503 


1-73905 


.59849 


1.67088 


.62245 


1.60657 


6 


55 


.55241 


1.81025 


.57541 


1.73788 


.59888 


1.66978 


.62285 


1-60553 


5 


56 


.55279 


1. 8090 1 


.57580 


1.73671 


.59928 


1.66867 


.62325 


1 .60449 


4 


57 


.55317 


1.80777 


.57619 


1.73555 


-59967 


1.66757 


.62366 


1.60345 


3 


58 


.55355 


1.80653 


.57657 


1.73438 


.60037 


1.66647 


.62406 


1.60241 


2 


59 


•55393 


1.80529 


.57696 


1.73321 


.60046 


1.66538 


.62446 


1. 60137 


I 


60 


•55431 


1 .80405 


.57735 


i^73205 


.60086 


1.66428 


.62487 


1 .60033 







Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 






6 


1° 


6 


0° 


5 


9° 


5 


8° 





MEASUREMENT OF RIGHT. TRIANGLES 



153 



32° 1 


33° 


34° 1 


3 


5° 




Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


/ 


.62487 


1.60033 


.64941 


1.53986 


•67451 


1.48256 


.70021 


1. 428 1 5 


60 


.62527 


1.59930 


.64982 


1.53888 


•67493 


1. 48163 


.70064 


1.42726 


59 


.62568 


1.59826 


.65023 


1.53791 


•67536 


1.48070 


.70107 


1.42638 


58 


.62608 


1.59723 


.65065 


1-53693 


.67578 


1-47977 


.70151 


1-42550 


57 


.62649 


1.59620 


.65106 


1.53595 


.67620 


1.47885 


i ^70194 


1.42462 


56 


.62689 


1. 595 1 7 


.65148 


1.53497 


.67663 


1.47792 


.70238 


1.42374 


55 


.62730 


1.59414 


.65189 


1.53400 


•67705 


1.47699 


.70281 


1.42286 


54 


.62770 


1.59311 


.65231 


1.53302 


•67748 


1.47607 


•70325 


1.42198 


53 


.62811 


1.59208 


■65272 


1.53205 


•67790 


1-47514 


•70368 


1.42110 


52 


.62852 


1.59105 


.65314 


1.53107 


•67832 


1.47422 


.70412 


1.42022 


51 


.62892 


1.59002 


•65355 


1.53010 


.67875 


1.47330 


.70455 


1.41934 


50 


.62933 


1.58900 


•65397 


1.52913 


.67917 


1.47238 


•70499 


1.41847 


49 


.62973 


1.58797 


•65438 


1.52816 


.67960 


1.47146 


•70542 


1.41759 


48 


.63014 


1.58695 


.65480 


1.52719 


.68002 


1.47053 


•70586 


1.41672 


47 


•6305s 


1.58593 


•65521 


1.52622 


.68045 


1.46962 


.70629 


1. 4 1 584 


46 


.63095 


1.58490 


•65563 


1-52525 


.68088 


1.46870 


•70673 


1.41497 


45 


.63136 


1.58388 


.65604 


1.52429 


.68130 


1.46778 


.70717 


1. 4 1 409 


44 


.63177 


1.58286 


.65646 


1.52332 


•68173 


1.46686 


.70760 


1.41322 


43 


.63217 


1.58184 


.65688 


1.52235 


.68215 


1.46595 


.70804 


1.41235 


42 


.63258 


1.58083 


.65729 


1.52139 


.68258 


1.46503 


.70848 


1.41148 


41 


.63299 


1.57981 


•65771 


1.52043 


.68301 


1.46411 


.70891 


1.41061 


40 


.63340 


1.57879 


.65813 


1.51946 


.6S343 


1.46320 


.70935 


1.40974 


39 


.63380 


1.57778 


.65854 


1-51850 


.68386 


1.46229 


.70979 


1.40887 


38 


.63421 


1.57676 


.65896 


1.51754 


.68429 


1.46137 


.71023 


1.40800 


37 


.63462 


1.57575 


.65938 


1.51658 


.68471 


1.46046 


.71066 


1.40714 


36 


.63503 


1.57474 


.65980 


1.51562 


.68514 


1-45955 


.71110 


1.40627 


35 


.63544 


1.57372 


.66021 


1. 5 1 466 


•68557 


1.45864 


.71154 


1.40540 


34 


.63584 


1.57271 


.66063 


1.51370 


.68600 


1-45773 


.71198 


1.40454 


33 


.63625 


1.57170 


.66105 


1.51275 


.68642 


1.45682 


.71242 


1.40367 


32 


.63666 


1.57069 


.66147 


1.51179 


.68685 


1.45592 


.71285 


1.40281 


31 


.63707 


1.56969 


.66189 


1. 5 1084 


.68728 


I-4550I 


.71329 


1.40195 


30 


.63748 


1.56868 


.66230 


1.50988 


.68771 


1.45410 


•71373 


1.40109 


29 


.63789 


1.56767 


.66272 


1.50893 


.68814 


1.45320 


.71417 


1.40022 


28 


.63830 


1,56667 


.66314 


1.50797 


.68857 


1.45229 


.71461 


1.39936 


27 


.63371 


1.56566 


.66356 


1.50702 


.68900 


1.45139 


•71505 


1.39850 


26 


.63912 


1.56466 


.66398 


1.50607 


.60942 


1.45049 


•71549 


1.39764 


25 


•63953 


1.56366 


.66440 


1.50512 


.68985 


1.44958 


•71593 


1.39679 


24 


.63994 


1.56265 


.66482 


1.50417 


.69028 


1.44868 


•71637 


1-39593 


23 


.64035 


1.56165 


.66524 


1.50322 


.69071 


1.44778 


.71681 


1.39507 


22 


.64076 


1.56065 


.66566 


1.50228 


.69114 


1.44688 


.71725 


1.39421 


21 


.64117 


1.55966 


.66608 


1.50133 


.69157 


1.44598 


•71769 


1.39336 


20 


.64158 


1.55866 


.66650 


1.50038 


.69200 


1.44508 


.71813 


1^39250 


19 


.64199 


1.55766 


•66692 


1.49944 


.69243 


1.44418 


.71857 


1.39165 


18 


.64240 


1.55666 


.66734 


1.49849 


.69286 


1.44329 


.71901 


1.39079 


17 


.64281 


1.55567 


.66776 


1.49755 


.69329 


1.44239 


.71946 


1.38994 


16 


.64322 


1.55467 


.66818 


1.49661 


.69372 


1.44149 


.71990 


1.38909 


15 


.64363 


1.55368 


.66860 


1.49566 


.69416 


1.44060 


.72034 


1.38824 


14 


.64404 


1.55269 


.66902 


1-49472 


.69459 


1.43970 


.72078 


1-38738 


13 


.64446 


1.55170 


.66944 


1.49378 


.69502 


1.43881 


.72122 


1-38653 


12 


.64487 


1.55071 


.66986 


1.49284 


•69545 


1.43792 


.72166 


1.38568 


11 


.64528 


1.54972 


.67028 


1. 49 1 90 


.69588 


1.43703 


.72211 


1.38484 


10 


.64569 


1.54873 


.67071 


1.49097 


.69631 


1.43614 


.72255 


1.38399 


9 


.64610 


1-54774 


.67113 


1 .49003 


.69675 


1.43525 


.72299 


1-38314 


8 


.64652 


1.54675 


.67155 


1 .48909 


.69718 


1-43436 


.72344 


1.38229 


7 


.64693 


1.54576 


.67197 


1.48816 


.69761 


1-43347 


.72388 


1-38145 


6 


.64734 


1.54478 


.67239 


1.48722 


.69804 


1-43258 


.72432 


1.38060 


5 


.64775 


1-54379 


.67282 


1.48629 


.69847 


1.43169 


-72477 


1-37976 


4 


.64817 


1.54281 


•67324 


1.48536 


.69891 


1.43080 


.72521 


1-37891 


3 


.64858 


1-54183 


.67366 


1.48442 


.69934 


1.42992 


.72565 


1-37807 


2 


.64899 


1.54085 


.67409 


1.48349 


.69977 


1.42903 


.72610 


1.37722 


I 


.64941 


1.53986 


.67451 


1.48256 


.70021 


1.42815 


.72654 


1-37638 





Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


f 


5 


7° 


5 


6° 


5 


5° 


5 


40 





154 



MECHANICS AND ALLIED SUBJECTS 





36° 


37° 


38° 


39° 




1 


Tan. 


Co-Tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


/ 


o 


.72654 


1-37638 


.75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


60 


I 


.72699 


1-37554 


.75401 


1.32624 


.78175 


1.27917 


.81027 


1.23416 


5Q 


2 


.72743 


1-37470 


.75447 


1-32544 


.78222 


1.27841 


.81075 


'^'2Z2>Al 


58 


3 


.72788 


1.37386 


.75492 


1-32464 


.78269 


1.27764 


.81123 


1.23270 


57 


4 


.72832 


1.37302 


•75538 


1-32384 


.78316 


1.27688 


.81171 


1.23196 


56 


5 


.72877 


1-37218 


•75584 


1.32304 


.78363 


1.27611 


.81220 


1.23123 


55 


6 


.72921 


1-37134 


.75629 


1.32224 


.78410 


1-27535 


.81268 


1.23050 


54 


7 


.72966 


1-37050 


•75675 


1.32144 


.78457 


1.27458 


.81316 


1.22^^ 


53 


8 


.73010 


1-36967 


.75721 


1.32064 


•78504 


1.27382 


.81364 


1.22904 


52 


9 


.73055 


1-36883 


.75767 


I -3 1 984 


.78551 


1.27306 


.81413 


I.2283I 


51 


ro 


.73100 


1.36800 


.75812 


1.31904 


.78598 


1.27230 


.81461 


1.22758 


50 


II 


.73^44 


1.36716 


.75858 


1-31825 


.78645 


1.27153 


.81510 


1.22685 


49 


12 


.73189 


1.36633 


.75904 


I-31745 


.78692 


1.27077 


.81558 


1.22612 


48 


13 


.73234 


1.36549 


.75950 


1. 3 1 666 


•78739 


1.27001 


.81606 


1.22539 


47 


14 


.73278 


1.36466 


.75996 


1.31586 


.78786 


1.26925 


.81655 


1.22467 


46 


IS 


.73323 


1.36383 


.76042 


1-31507 


.78834 


1.26849 


.81703 


1.22394 


45 


i6 


.73368 


1.36300 


.76088 


1.31427 


.78881 


1.26774 


.81752 


1. 22321 


44 


17 


.73413 


1.36217 


.76134 


1.31348 


.78928 


1.26698 


.81800 


1.22249 


43 


i8 


.73457 


1.36133 


.76180 


1. 3 1 269 


.78975 


1.26622 


.81849 


1.22176 


42 


19 


.73502 


1.36051 


.76226 


1.31190 


.79022 


1.26546 


.81898 


1. 22 104 


41 


20 


.73547 


1.35968 


.76272 


1.31110 


.79070 


1.26471 


.81946 


I.22O3I 


40 


21 


.73592 


1.35885 


.76318 


1.31031 


.79117 


1.26395 


.81995 


1.21959 


39 


22 


.73637 


1-35802 


.76364 


1.30952 


.79164 


1.26319 


.82044 


1.21886 


38 


23 


.73681 


1.35719 


.76410 


1.30873 


.79212 


1.26244 


.82092 


I.21814 


37 


24 


.73726 


1.35637 


.76456 


1.30795 


.79259 


1. 26169 


.82141 


I.21742 


36 


25 


.73771 


1.35554 


.76502 


1. 307 16 


.79306 


1.26093 


.82190 


1.21670 


35 


26 


.73816 


1.35472 


.76548 


1.30637 


.79354 


1.26018 


.82238 


1.21598 


34 


27 


.73861 


1.35389 


.76594 


1.30558 


.79401 


1.25943 


.82287 


I.21526 


33 


28 


.73906 


1.35307 


.76640 


1.30480 


•79449 


1.25867 


.82336 


I.2I454 


Z2 


29 


.73951 


1.35224 


.76686 


1. 30401 


.79496 


1.25792 


.82385 


I.21382 


31 


30 


.73996 


1.35142 


.76733 


1.30323 


.79544 


1.25717 


.82434 


I.213IO 


30 


31 


.74041 


1.35060 


•76779 


1.30244 


.79591 


1.25642 


.82483 


I.21238 


29 


32 


.74086 


1.34978 


.76825 


1.30166 


.79639 


1.25567 


.82531 


I.2I166 


28 


33 


.74131 


1.34896 


.76871 


1.30087 


.79686 


1.25492 


.82580 


1. 2 1094 


27 


34 


.74176 


1. 34814 


.76918 


1.30009 


•79734 


1-25417 


.82629 


1.21023 


26 


35 


.74221 


1.34732 


.76964 


1.29931 


•79781 


1.25343 


.82678 


I.2095I 


25 


36 


.74267 


1.34650 


.77010 


1-29853 


•79829 


1.25268 


.82727 


1.20879 


24 


37 


.74312 


1.34568 


'77057 


1.29775 


.79877 


1.25193 


.82776 


1.20808 


2Z 


38 


.74357 


1.34487 


.77103 


1.29696 


.79924 


1.25118 


.82825 


1.20736 


22 


39 


.74402 


1.34405 


•77149 


1.29618 


.79972 


1-25044 


.82874 


1.20665 


21 


40 


.74447 


1.34323 


.77196 


1.29541 


.80020 


1.24969 


.82923 


1-20593 


20 


41 


.74492 


1.34242 


.77242 


1.29463 


.80067 


1.24895 


.82972 


1.20522 


19 


42 


.74538 


1.34160 


.77289 


1.29385 


.80115 


1.24820 


.83022 


I.2045I 


18 


43 


.74583 


1.34079 


.77335 


1.29307 


.80163 


1.24746 


.83071 


1.20379 


17 


44 


.74628 


1.33998 


.77382 


1.29229 


.80211 


1.24672 


.83120 


1.20308 


16 


45 


.74674 


1.33916 


.77428 


i.:^9i52 


.80258 


1.24597 


.83169 


1.20237 


15 


46 


.74719 


1-33835 


•77475 


1.29074 


.80306 


1.24523 


.83218 


1.20166 


14 


47 


.74764 


1.33754 


.77521 


1.28997 


.80354 


1.24449 


.83268 


1.20095 


13 


48 


.74810 


1.33673 


.77568 


1.28919 


.80402 


1.24375 


.83317 


1.20024 


12 


49 


.74855 


1.33592 


.77615 


1.28842 


.80450 


1.24301 


.83366 


1-19953 


II 


50 


.74900 


1.33511 


.77661 


1.28764 


.80498 


1.24227 


.83415 


1.19882 


10 


51 


.74946 


1.33430 


.77708 


1.28687 


.80546 


1.24153 


.83465 


1. 19811 


9 


52 


.74991 


1.33349 


.77754 


1.28610 


.80594 


1.24079 


.83514 


1. 19740 


8 


53 


.75037 


1.33268 


.77801 


1.28533 


.80642 


1.24005 


.83564 


1. 19669 


7 


54 


.75082 


1.33187 


.77848 


1.28456 


.80690 


1.23931 


.83613 


1.19599 


6 


55 


.75128 


1.33107 


.77895 


1.28379 


.80738 


1-23858 


.83662 


I. 19528 


5 


56 


.75173 


1.33026 


-77941 


1.28302 


.80786 


1-23784 


.83712 


1-19457 


4 


57 


.75219 


1.32946 


-77988 


1.2822s 


.80834 


1-23710 


.83761 


1-19387 


3 


58 


.75264 


1.32865 


-7S035 


1.28148 


.80882 


1.23637 


.83811 


1.19316 


2 


59 


•75310 


1.32785 


.78082 


1.28071 


.80930 


1.23563 


.83860 


1.19246 


I 


60 


•75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


.83910 


I. 19175 





/ 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan 


Tan. 


Co-tan. 


Tan 


t 




5, 


30 


5^ 


2° 


5 


1° 


51 


0° 





MEASUREMENT OF RIGHT TRIANGLES 



155 





40° 


i 4P 


42° 


43° 




/ 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


f 


o 


.83910 


1.19175 


.86929 


1.15037 


.90040 


i.iro6i 


•93252 


1.07237 


6c 


I 


.83960 


1.19105 


.86980 


1. 14969 


.90093 


1. 1 0996 


.93306 


1.07174 


5Q 


2 


.84009 


1.19035 


.87031 


1. 1 4902 


.90146 


1.10931 


•93360 


1.07 1 12 


58 


3 


.84059 


1. 1 8964 


.87082 


1. 14834 


.90199 


1. 1 0867 


•93415 


1.07049 


57 


4 


.84108 


1.18894 


.87133 


1. 14767 


.90251 


1. 10802 


•93469 


1.06987 


56 


s 


.84158 


1. 18824 


.87184 


1. 1 4699 


•9^304 


1.10737 


•93524 


1.06925 


55 


6 


.84208 


1. 18754 


.87236 


1. 14632 


•90357 


1.10672 


•93578 


1.06862 


54 


7 


.84258 


1. 1 8684 


.87287 


1. 14565 


.90410 


1. 10607 


•93633 


1.06800 


53 


8 


.84307 


1.18614 


.87338 


1. 14498 


.90463 


1.10543 


.93688 


1.06738 


52 


9 


.84357 


1. 18544 


.87389 


1. 14430 


.90516 


I. 10478 


•93742 


1.06676 


51 


lo 


.84407 


1. 18474 


.87441 


1.14363 


.90569 


1.10414 


•93797 


1. 066 13 


50 


II 


•84457 


1. 1 8404 


.87492 


I. 14296 


.90621 


1. 10349 


.93852 


1. 065 5 1 


49 


12 


.84507 


1. 18334 


.87543 


1.14229 


.90674 


1. 10285 


.93906 


1.06489 


48 


13 


.84556 


1. 18264 


•87595 


1.14162 


.90727 


1. 10220 


.93961 


1.06427 


47 


14 


.84606 


1.18194 


.87646 


1. 14095 


.90781 


1.10156 


.94016 


1.06365 


46 


15 


.84656 


1.18125 


.87698 


1. 14028 


.90834 


1.10091 


.94071 


1.06303 


45 


i6 


.84706 


1. 18055 


.87749 


1.13961 


.90887 


1. 10027 


.94125 


1.06241 


44 


t7 


.84756 


I.I 7986 


.87801 


1. 13894 


.90940 


1.09963 


.94180 


1. 06 1 79 


43 


i8 


.84806 


1^7916 


.87852 


1.13828 


•90993 


1 .09899 


.94235 


1.06117 


42 


19 


.84856 


1. 1 7846 


.87904 


1.13761 


.91046 


1.09834 


.94290 


1.06056 


41 


20 


.84906 


1. 17777 


.87955 


1. 1 3694 


.91099 


1.09770 


.94345 


1.05994 


40 


JI 


•84956 


1*17708 


.88007 


1. 13627 


•91153 


1 .09706 


.94400 


1.05932 


39 


J2 


.85006 


1.17638 i 


.88059 


1.13561 


.91206 


1.09642 


•94455 


1.05870 


38 


*3 


.85057 


1.17569 


.88110 


1. 1 3494 


.91259 


1.09578 


•94510 


1.05809 


37 


'4 


.85107 


1.17500 


.88162 


1. 13428 


•91313 


1. 095 1 4 


.94565 


1^05747 


36 


5 


.85157 


• 1. 1 7430 


.88214 


1.13361 


•91366 


1.09450 


1 .94620 


1.05685 


35 


»6 


.85207 


1.17361 


.88265 


1.13295 


.91419 


1.09386 


' .94676 


1.05624 


34 


7 


.85257 


1. 17292 


.88317 


1. 13228 


.91473 


1.09322 


.94731 


1.05562 


33 


>8 


.85307 


1. 17223 


.88369 


1.13162 


•91526 


1.09258 


.94786 


1. 05501 


32 


*9 


.85358 


1.17154 


.88421 


1. 1 3096 


.91580 


1.09195 


.94841 


1^05439 


31 


o 


.85408 


1. 1 7085 


.88473 


1. 13029 


•91633 


1.09131 


.94896 


1^05378 - 


30 


I 


.85458 


1.17016 


.88524 


1.12963 


.91687 


1.09067 


.94952 


1^05317 


29 


2 


.85509 


1.16947 


•88576 


1. 12897 


.91740 


1.09003 


.95007 


1.05255 


28 


3 


.85559 


I. 16878 


.88628 


1.12831 


•91794 


1.08940 


.95062 


1.05194 


27 


4 


.85609 


I. I 6809 


.88680 


1.12765 


•91847 


1.08876 


.95118 


1-05133 


26 


5 


.85660 


1.16741 


.88732 


1. 12699 


.91901 


1. 088 1 3 


•95173 


1.05072 


25 


6 


.85710 


1. 16672 


.88784 


1.12633 


•91955 


1.08749 


.95229 


1.05010 


24 


7 


.85761 


1. 1 6603 


.88836 


1. 12567 


.92008 


1.08686 


.95284 


1.04949 


23 


8 


.85811 


1. 16535 


.88888 


1.12501 


.92062 


1.08622 


.95340 


1.04888 


22 


9 


.85862 


1. 16466 


.88940 


I.I2435 


.92116 


1.08559 


.95395 


1.04827 


21 


^o 


.85912 


1. 16398 


.88992 


1. 12369 


.92170 


1 .08496 


•95451 


1.04766 


20 


I 


.85963 


1. 16329 


.89045 


1. 12303 


.92224 


1.08432 


•95506 


1.04705 


19 


2 


.86014 


1.16261 


.89097 


1.12238 


.92277 


1.08369 


•95562 


1.04644 


18 


3 


.86064 


1.16192 


.89149 


1.12172 


.92331 


1 .08306 


•95618 


1^04583 


17 


\4 


.86115 


1.16124 


.89201 


1.12106 


.92385 


1.08243 


•95673 


1.04522 


16 


5 


.86166 


I. 16056 


•89253 


1.12041 


.92439 


1. 08 1 79 


•95729 


1. 0446 1 


15 


^6 


.86216 


1. 1 5987 


.89306 


1-11975 


•92493 


1.08116 


•95785 


1. 0440 1 


14 


^7 


.86267 


1.15919 


•89358 


1.1T909 


•92547 


1.08053 


.95841 


1 .04340 


13 


f8 


.86318 


1.15851 


•89410 


1.11844 


.92601 


1.07990 


•95897 


1.04279 


12 


\9 


.86368 


1.15783 


•89463 


1.11778 


.92655 


1.07927 


•95952 


1.04218 


II 


'O 


.86419 


1.15715 


.89515 


1.11713 


.92709 


1.07864 


.96008 


1. 04 1 58 


10 


I 


.86470 


1. 15647 


•89567 


1.11648 


•92763 


1.0780T 


.96064 


1.04097 


9 


'2 


.86521 


1.15579 


.89620 


1.11582 


.92817 


1.07738 


.96 1 20 


1.04036 


8 


'3 


.86572 


1.15511 


.89672 


1.11517 


.92872 


1.07676 


.96176 


1.03976 


7 


'4 


.86623 


1. 1 5443 


.89725 


1.11452 


.92926 


1.07613 


.96232 


I -03915 


6 


'5 


.86674 


1. 15375 


.89777 


1.11387 


.92980 


1.07550 


.96288 


1. 0385 "5 


5 


;6 


.86725 


1. 15308 


.89830 


1.11321 


•93034 


1.07487 


.96344 


1.03794 


4 


'7 


.86776 


1. 15240 


.89883 


1.11256 


.93088 


1.07425 


.96400 


1.03734 


3 


;8 


.86827 


1. 15172 


•89935 


1.11191 


.93143 


1.07362 


•96457 


1.03674 


2 


J9 


.86878 


1.15104 


.89988 


1.11126 


.93197 


1.07299 


•96513 


1. 0361 3 


1 


K) 


.86929 


I. 15037 


.90040 


1.11061 


•93252 


1.07237 


•96569 


1-03553 





/ 


Co-tan 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


/ 




^ 4( 


r 


4< 


go 


4 


7° 


4 


3° 





156 



MECHANICS AND ALLIED SUBJECTS 





44° 






440 






440 




/ 


Tan. 


Co-TAN. 


/ 


/ 


Tan. 


Co-TAN. 


30 


f 

41 


Tan. 


Co-TAN. 


/ 


o 


.96569 


1-03553 


60 


21 


97756 


1.02295 


.98901 


I.OIII2 


ig 


1 


.96625 


1^03493 


59 


22 


97813 


1.02236 


38 


42 


.98958 


I.OIO53 


18 


2 


.^6681 


I -03433 


58 


23 


97870 


1.02 1 76 


37 


43 


.99016 


1.00994 


17 


3 


•96738 


1.03372 


57 


24 


97927 


1. 021 17 


36 


44 


.99073 


1.00935 


16 


4 


■96794 


1.033 1 2 


56 


25 


97984 


1.02057 


35 


45 


.99131 


1.00876 


15 


5 


.96850 


1.03252 


55 


26 


98041 


1. 01 998 


34 


46 


.99189 


1. 008 1 8 


14 


6 


.96907 


1.03 192 


54 


27 


98098 


I.OI939 


35 


47 


.99247 


1.00759 


13 


7 


.96963 


1.03132 


53 


28 


98155 


I.O1879 


32 


48 


•99304 


1. 00 70 1 


12 


8 


.97020 


1.03072 


52 


29 


98213 


1.01820 


31 


49 


.99362 


1.00642 


II 


9 


.97076 


1. 030 1 2 


51 


30 


98270 


I.OI761 


30 


50 


.99420 


1.00583 


10 


lO 


•97133 


1.02952 


50 


31 


98327 


1. 01 702 


29 


51 


•99478 


1.00525 


9 


II 


.97189 


1.02892 


49 


32 


98384 


1. 01 642 


28 


52 


•99536 


1.00467 


8 


12 


.97246 


1.02832 


48 


33 


98441 


I.OI583 


27 


53 


.99594 


1 .00408 


7 


13 


•97302 


1.02772 


47 


34 


98499 


1.01524 


26 


54 


.99652 


1.00350 


6 


14 


•97359 


1.02713 


46 


35 


98556 


I.OI465 


25 


55 


.99710 


1.00291 


5 


15 


•97416 


1.02653 


45 


36 


98613 


1. 01 406 


24 


56 


.99768 


1.00233 


4 


i6 


•97472 


1.02593 


44 


37 


98671 


I.OI347 


23 


57 


.99826 


1. 001 75 


3 


17 


•97529 


1.02533 


43 


38 


98728 


1. 01 288 


22 


58 


.99884 


1.00116 


e 


i8 


.97586 


1.02474 


42 


39 


98786 


I.OI229 


21 


59 


.99942 


1.00058 


I 


19 


•97643 


1.02414 


41 


40 


98843 


I.OII70 


20 


60 


I 


I 





2Q 


.97700 


1.02355 


40 








/ 


f 









/ 


Co-TAN. 


Tan. 


/ 


' c 


'o-tan. 


Tan. 


Co-TAN. 


Tan. 


/ 




4 


5° 






4 


5° 






45° 





6. Complete the following table using the table of sines 



cosines, etc. 










1. Sin 30° 


9. Cot 


48° 




2. Cos 60° 


10. Tan 


42° 




3. Sin 20° 10' 


11. Cos 


27° 




4. Cos 69° 50' 


12. Sin 


63° 




5. Cos 36° 


13. Tan 


45° 




6. Sin 54° 


14. Cot 


45° 




7. Tan 12° 


15. Sin 


20° 




8. Cot 78° 


16. Cos 


70° 



From the above table what do you find about the sin 
30° and the cos 60^ [that is, the cos of (90° - 30°)], also 
about the sin 20° 10' and the eos of (90° - 20° 10'), 
that is, the cos 69° 50'; about the tan 12° and the cot 
of (90° - 12°) or cot 78°. In other words, this is true. 

The sine of any angle less than 90° equals the co-func- 
tion or cosine of (90° minus that angle), also, the tangent 
of any angle less than 90°, for example, the tangent of 
12° equals the co-function or cotangent of (90° — 12°) 



MEASUREMENT OF RIGHT TRIANGLES 



157 



or equals the cot of 78°. In the same way the sine 20° 
10' = COS. 69° 50', the cot 48° = tan (90° - 48°) or tan 
42° the cos 30° = sin. (90° - 30°) or sin 60°, etc. 
This is a good rule to remember. 

62. Line Values of Functions. — The values of the sine, 
cosine, tangent and cotangent 
of angles can be represented 
by the lengths of lines as fol- 
lows: In Fig. 109 the circle 
with center is drawn with a 
radius (one-half the diameter) 
equal to 1 in. CD is a horizon- 
tal and EF a vertical diameter. 
The radius OA makes any con- 
venient angle. ^^a^' with ra- 
dius OC and from point A line 
A 5 is drawn at right angles to line OC making the an- 
gle at B equal to 90° and giving the right triangle OAB. 

From the definition of the sine ( = , -. ) we 

\ hypothenuse/ 

have in the right triangle OAB, The sine of angle 

a = , yr-r' But OA = 1 by construction (since we 

hyp. OA , -^ "^ 

made the circle 

side AB 




a = 



1 



with OA = 1 in.). Therefore sine 
or side AB. Hence the sine of a is equal 



to the length of the line AB. In the figure, AB measures 
.5 of an inch. Therefore sine a = .5. Notice that if 
angle a were smaller the line AB would be smaller, that 
is, sine a would be smaller. If angle a were larger , line 
AB would be larger and therefore sine a would be larger. 
When angle a = 0°, line AB = in. and sine a = 0, 
that is, sine 0° - 0. When a = 90°, line AB = line 
OE in length or 1 in. and therefore sine 90° = 1. There- 



158 



MECHANICS AND ALLIED SUBJECTS 



fore for angle between 0° and 90° the sine increases when 
the angle increases. 

Referring again to Fig. 109, in the right triangle OAB 

adjacent side side 05 



the cosine 
side OB 



of angle a 



hypothenuse hyp. OA 



1 

That 
angle 



or side OB. 

is, the line OB 
a. When angle 




equals to scale the cosine of 
a = 0, OB = OC, and cos 
a = 1. When angle a = 90°, line 
OB = 0, and cosine 90° = 0. 

Therefore, for angles between 
0° and 90° the cosine decreases 
as the angle increases. The fig- 
ure on this sheet is like the pre- 
ceding figure, page 157 drawn 
again here for convenience. Re- 
ferring to right triangle OAB, 
the tangent of angle a = 
opposite side _ side AB 
adjacent side 



Notice 



side OB 

that as angle a increases, side AC (that is, the nu- 
merator of the fraction which is the tangent) increases, 
and at the same time side OB, the denominator of the 
fraction, decreases. Therefore as angle a increases the 
tangent increases and also if angle a decreases the tangent 
of angle a decreases. 

In right triangle OAB the cotangent of angle a = 
adjacent side _ side OB 
opposite side 
side 



Then as angle a increases. 



side AB 
OB decreases and side AB increases; therefore the 
fraction representing the cotangent decreases. As angle a 
decreases side OB increases arid side AB decreases, and the 
value of the fraction increases, that is, the cotangent 



MEASUREMENT OF RIGHT TRIANGLES 



159 



increases. The following table for values of 0° and 90° 
angles should be carefully studied and understood from 
Fig. 110. 



Vngle 


Sine 


Cosine 


Tangent 


Cotangent 


0° 





1 





Infinity 


90° 


1 





Infinity 






63. Values of the Sine, Cosine, Tangent, and Co- 
tangent for the Common Angles. — In a 30°-60° right 





Fig. 112. 



triangle shown in Fig. Ill the sides always have a ratio 
of 1, 2, and \/3 as given. You can prove this for your- 
selves by carefully constructing such a triangle on the 
drawing board and measuring the lengths of the sides. 
Therefore from our definition of the sine, cosine, etc., of 
an angle we have 



Sin 30° 

Cos 30° 
Tan 30° 



opp. side 

hyp- 

adj. side 

hyp. 
opp. side 



= X = .5 



2 

V3 
2 
1 



adj. side ^3 



= .866 



= .577 



160 MECHANICS AND ALLIED SUBJECTS 

Cot 30° = ^^4^ = ^ = 1.732 
opp. side 1 

Sin 60° = ^^^ = y^ = .866 
hyp. 2 

^ ^^^ adi. side 1 

Cos 60° = -— = - = .5 

hyp. 2 

Tan 60° = ^-^^ = ^ = 1.732 
adj. side 1 

adj. side 1 

Cot 60 = ri- == — 7:^ = .577 

opp. side V3 

In a 45°-45° right triangle as shown in Fig. 112 the sides 
always have a ratio of 1, 1 and \/2 as shown. Therefore 
for this figure: 

Sin 45° = ^^^^ = ^ = .707 
hyp- V2 

Cos 45° = ^--^ = 4= = .707 
hyp- V2 

Tan 45° = ^^^1^ = \ = 1.00 
adj. side 1 

Cot 45° = "-^y^ = J = 1.00 
opp. side 1 

These values are tabulated below for reference. 



Angle 


Sine 


Cosine 


Tangent 


Cotangent 


30° 


.5 


.866 


.577 


1.732 


45° 


.707 


.707 


1.00 


1.00 


60° 


.866 


.5 


1 . 732 


.577 



64. Methods of Working out Right Triangles. — In 

working out problems by the use of the sine, cosine, etc., 
we use one or the other, according to the sides of the 
triangle which are to be found and the sides which are 



MEASUREMENT OF RIGHT TRIANGLES 



161 



known. For example in Fig. 113 the hyp. = 10 in. and 
the smaller acute angle = 30°. We want to find side 
A, side B and angle h. In any right triangle the sum of 
the two acute angles equals 90°, therefore angle h + 
30° = 90° and angle 6 = 90° - 30° or 60°. To find 

opp. side side A 



side A we have sine 30° = 
side A 



If 



sme 



30° = 



10 



hyp. 10 in. 

, side A = 10 X sine 30° (sine 30° = .5), 



^0' f 


A 


or 5 

side 




5 
Fig. 113. 

therefore side A = I 
we use the cosine as f 

cos 30° 


X .5 

qIIows : 

adj. 


5 
Fig. 114.. 

in. To find sic 
side B 


hyp. 


~ 10 



therefore side B = 10 X cos 30°, (cos 30° = .866), 
therefore side B = 10 X .866 or 8.66 in. 

From this explanation just given and using Fig. 114 
complete the following table. The first set of values 
have been worked out as a guide. 





Angle a 


Angle b 


Side C 


Side A 


Side B 


1 


30° 


60° 


16 


8 


13.9 


2 


30" 




24 






3 




30° 


38 






4 


45° 




7.5 






5 


60° 




62 






6 




75° 


35 







11 



162 



MECHANICS AND ALLIED SUBJECTS 



Draw a separate triangle for each set of values and put 
the lengths of sides and sizes of angles on each figure as 
well as putting them in the table. 

If as in the triangle shown (Fig. 115) we had given 
angle a = 30° and side J? = 8, to find side A we use the 

side A 
tanget as follows: tan 30° = -7-3 — ^' therefore side A = 
^ side B 

side B X tan 30° or side A = 8 X .577 = 4.62. 

If as in the triangle shown in Fig. 116 we have the side 





A = 10 and angle a = 24° and wish to find side B we 
use the cotangent as follows: 

side B 



cot 24^ 



10 



Therefore side 5 = 10 X cot 24° or side B = 10 X 
2.25 or 22.5. From the explanation just given and 
using Fig. 117 complete the following table. 





Angle a 


Angle b 


Side A 


Side B 


1 




29° 


16 




2 


80° 






8.4 


3 




36° 


35 




4 


15° 






45 


5 




75° 


7.5 




6 


60° 






24 



MEASUREMENT OF RIGHT TRIANGLES 163 



When the base B and angle a are given (Fig. 118) to 
find the hypothenuse proceed as follows: 

side B 



or 



cos 40° = 
cos 40° = 



hyp. 
20 



hyp. 





Therefore cos 40° X hyp. = 20 and from this the hyp. = 

20 20 



cos 40° .766 



26.1 



When the side A and angle a are given (Fig. 119) to 
find the hypothenuse proceed as follows: 





3=16 
Fig. 120. 



.^o side ^ . ^ 14 

sni 16° = -r or sin 16° = ^^^ 

hyp. hyp. 

Therefore hyp. X sin 16° = 14 and hyp. = ~,^ 

^^^p- = :276 = ^0-^ 



14 



sin 16° 



or 



164 MECHANICS AND ALLIED SUBJECTS 

7. From the explanation just given fill in the following table: 



' 


Angle a 


Angle h 


Side A 


S'ldeB 


Hyp. 


1 


12° 




10 






2 




16° 30' 




20 




3 


43° 




15 






4 




27° 10' 




63 




5 


36° 




23.8 






6 




32° 




42 





When we have two sides of a right triangle and want 

to find the angles we proceed as follows: In the triangle 

shown in Fig. 120 A = 12, B = 16, hyp. = 20. To 

find angles a and 6. 

opp. si de 12 3 

bm a = — r = 7^ = c = -6. It sm a = .6 then 

hyp. 20 5 

angle a is one whose sine is .6. We therefore look in the 
tables for an angle whose sine is .6 and on page 142 find 
the number .59995 which is the nearest one to .60000; 
the angle is therefore 36° 52'. 

We can find angle a also from the rule 
_ adj. s ide _ 16 _ 8 
^^^ ^ " ~hy^r^ ~ 20 " lO 
Looking in the tables for the angle whose cosine is .80000 
we find on page 142 the angle 36° 52'. 

We can find angle a also from the tangent rule as 
follows : 

opp. side 12 3 
~ l6 ~ ~ 



= .8 



Tan a = 



.75 



adj. side 16 4 

and therefore (page 154) angle a = 36° 52', or from the 

cotangent rule 

16 4 
cot a = ^^ = ^ = 1.33 

and from the tables (page 154) angle a = 36° 52'. 



MEASUREMENT OF RIGHT TRIANGLES 



165 



Having obtained angle a from any one of the rules 
just given, angle b equals 90° — angle a, since angle a + 
angle b = 90°. 

8. From the explanation just given and using Fig. 121, find the 
value of the sine of angle a and the value of angle a, completing the 
following table. 







Side A 


Hyp. 


Sine a Angle a 


1 


25 


50 






2 


7 


35 






3 


28 


38 






4 


15 


20 






5 


12 


32 







9. Using Fig. 122, find the value of the cosine of angle a and 
angle a, completing the following table. 





Side B 


Hyp. 


Cosine a 


Angle a 


1 


10 


20 






2 


3 


4 






3 


30 


75 






4 


16 


28 






5 


14 


32 







10. For Fig. 123, find the tangent of angle a and the value of 
angle a, completing the following table. 



166 



MECHANICS AND ALLIED SUBJECTS 





Side A 


Side B 


Tangent a 


Angle a 


1 


10 


24 






2 


36 


70 






3 


3 


8 






4 


56 


40 






5 


38 


112 









11. For Fig. 124, find the cotangent of angle a and the value of 
angle a, completing the following table. 





Side A 


Side 5 

1 


Cotangent a 


Angle a 


1 


7 


32 






2 


27 


'38 






3 


14 


18 






4 


12 


16 






5 


23 


42 








12. Using Fig. 125, complete the following table, drawing a 
separate triangle for each problem. 



MEASUREMENT OF RIGHT TRIANGLES 



167 





Angle a 


Angle b 


8ide A 


Side B 


Side C 


1 


60° 








75 


2 




40° 




35 




3 




32° 


15 






4 


75^^ 




20 






5 


15° 






30 




6 






25 




50 



65. Applications of Triangular Functions. — Following 
are applications of the preceding rules to the solution of 
problems in daily shop work. 

13. To find the depth of a F thread. Fig. 126 shows a section 
of the thread. To find the depth of the thread or distance CD 



\- Pitch 





%-Wlb. 



Fig. 126. 



Fig. 127. 



in triangle DBC. Side CD = side CB X cos 30° = pitch X .866. 
The pitch in in. always equals th^eads^peTTn. 

Find the depth of V threads as follows: 8 per in., 20 per in., 
12 per in., and 40 per in. 

14. In a right triangle the hypothenuse is 20 in. and the height 
is to be twice the base. Find the height, base, and all the angles 
of the triangle. (Note. — Let the base be x in. and the height 

2x 

2x in. The tangent of the base angle will then be — or 2.) 

x 

15. Fig. 127 shows a first-class lever. If W2 = 40 lb. to find 



168 



MECHANICS AND ALLIED SUBJECTS 



Wi we have the rule 40 X 8 = TFi X distance AB, but length 
AB = U X cos 15° = 14 X .966 = 13.5, therefore 40 X 8 = Wi 

X 13.5 and Wi = ^ 



13.5 



23.7 lb. Arts. 



16. With a first-class lever as in Fig. 128, find Wi necessary to 
balance Wo with lever arms as shown. 




^Yf;^i 



Wo^mn 



Fig. 128. 




Yf^IO-lb.Cl 
Fig. 129 



I2'''30' 



17. As per Fig. 129, find the force necessary at F to balance the 
weight of 10 lb. shown. 




F[±2W^20Ik 



Fig. 130. 

18. As shown in Fig. 130, find the force necessary at /^ to balance 
the 20 lb. What class of lever is this one? 

19. What weight in lb. at W (Fig. 131) is required to set the 
safety valve so as to lift at 200 lb. per sq. in. pressure with a 2- 
in. diam. valve? 

20. As per Fig. 132, what size valve is required with steam at 
150 lb. per sq. in. to lift the weight of 80 lb. at TF, with the lever 
arms as shown? 

21. In a laying out job the distance CB Fig. 133 is taken as 16 
in., the angle a = 24°. Find the distance between centers A and 
B. 

22. The tangent of an angle is .45573. What are its sine, 
cotangent, and cosine?. 



MEASUREMENT OF RIGHT TRIANGLES 



169 



23. For the wedge shown (Fig. 134) find the angle between its 
sides. 

24. Find the angle between the sides of a wedge 1 ft. long, 
% in. wide at one end and 3^^ in. wide at the other end. 

25. The sides of a wrought-iron wedge form an angle of 2° 23'. 
Find the taper (decrease in thickness) per ft. of length. 




t)/r 




□ yy- 



W=dO-/^ 



Fig. 132. 



26. A reamer 1% in. at the small end is 16 in. long and has a 
taper of % in. in. 12 in. 0i in. increase in diam. per ft.). What is 
the diameter at the large end and what is the angle between the 
two sides? 

27. Find the angle which the sides of a wedge make with one 
another for a taper of % in. per ft. 



c^ 


■"a^" 


0^^ 

90 


k- 


/6"-- 


H 



3 



Fig. 133. 




Fig. 134. 



28. Fig. 135 shows a telegraph pole with a guy wire attached 
20 ft. up on pole and anchored 10 ft. from base of pole. Find 
length of wire required. Allow 2 ft. extra for fastening wire. 
(Note. — Find first angle a, then get side C from the sine or cosine 
rule.) 

29. Find the angle of incline for a grade of 3 per cent., and the 
number of feet rise in a distance of 2 miles. (A 1 per cent, grade 
is one which rises 1 ft. vertically for every 100 ft. horizontally.) 



170 



MECHANICS AND ALLIED SUBJECTS 



How many foot-pounds of work are required to lift a 600-ton 
train up this distance against gravity? 

30. Find the average angle of incline for the roadbed from 
Altoona, Pa., toGallitzin, Pa., the rise being about 980 ft. in 10 miles. 

31. Two adjacent spokes in a wheel make with one another an 
angle whose cosine is .91355. What is the angle? How many 
spokes are there in the wheel? 

32. A belt passes over two pulleys whose centers are 11 ft. 





Fig. 136. 




Fig. 135. 



Fig. 137. 



apart. The pulley diameters are 18 in. and 42 in. Find the angle 
between the two sides of the belt. 

33. Find the angle between the sides of a belt connecting two 
pulleys 24 in. and 12 in. in diam. and with centers 6 ft. apart. 

34. Find the length of a side of an equilateral triangle inscribed 
in a 24-in. diam. circle. Fig. 136 shows the circle and triangle, 
Drawing the radii OA and OB we have the triangle OAB. Line 
OC divides this triangle into two equal right triangles OAC and OCB. 



MEASUREMENT OF RIGHT TRIANGLES 171 

Angle AOC = 60°. To find side AB we find AC first, then AB = 
2 X AC. Now in triangle AOC 

AC 

YYJ = sine 60° 

Therefore AC = OA X sin 60° = OA X .866 

But OA is the radius of the circle or 12 in. 

Therefore AC = 12 X .866 = 10.4 in. and AB = 2 X AC = 2 X 

10.4 = 20.8 in. Ans. 

35. Find the length of one side of each of the following equal- 
sided inscribed figures. 

xT.,»^u^.. ^f „i „:^^^ Diain of circle in which figure 

Number of equal sides -^ inscribed 

5 18 in. 

7 24 in. 

36. As shown in Fig. 137, find the distance AB to be laid off for 
drilling two of eight equally spaced holes on an 8-in. diam. circle. 

angle «=^°= 45" 

angle h = -^ = 223^°. 

Therefore in right triangle BCD 

side BD . , . , ^ 

'A rC ~ ^^^ angle 6, therefore 

side BD = side BC X sin h. 

= 4 in. X sin 22>^°. 

= 4 X .383 = 1.532 in. 
and AB = 2 X BD = 2 X 1.53 = 3.06 in. Ans. 

37. Find the distance at which to set dividers for laying off 12 
equally spaced holes on a 42-in. diam. circle. 

38. What is the distance in in. between any two consecutive 
centers when 17 are spaced an equal distance apart on a 1-ft. 
diam. circle? 

39. If the straight distance between two consecutive centers 
on a 12-in. diam. circle is 1}^ in., what is the angle formed at the 
center of the circle by the radii drawn to these two centers? 

40. In the sketch. Fig. 138, showing a cross-sectional view of a 



172 



MECHANICS AND ALLIED SUBJECTS 



boiler shell 72-in. diam. with water line 20 in. above the center of 
shell, find the length of the water line "x'' in in. 

41. What is the taper per ft. of length for a wedge, the sides of 
which form an angle whose tangent is .0626? 

42. In Fig. 139, diameter AB = 2 in., chords AC and AD are 





Fig. 138. 



Fig. 139. 



each }/2 ill- and angles C and D are each 90°. Find angle a and 
angle h. Angle b is that used in constructing the ''acme'' thread. 
Remember how this figure is drawn, to get this angle for an ''acme 
thread." 

43. In Fig. 140, find the length of the center lines of rafters A 





Fig. 140. 



Fig. 141. 



6 2 

and B, (Note. — ^The tangent of angle a = Tc = k = .4. Find 

angle a in the tables. Then knowing a and the 15-ft. side of the 
triangle CDF find side A. In the same w^ay to find side B, find 
angle b first, then work out the right triangle CDE for side B 
knov/ing angle b and the 10-ft. side. 

44. A line shaft is 18 ft. above and 15 ft. to the left of a motor 



MEASUREMENT OF RIGHT TRIANGLES 



173 



shaft. Find the distance between these shafts by drawing the 
right triangle, using the tables to get one of the acute angles and 
then the sine or cosine rule to get the hypothenuse of the triangle 
or the answer. 

45. Find the angle between the belt on two sides of pulleys 
18 in. and 42 in. in diam. and centers 11 ft. apart. Make a 
sketch of the belt and pulleys. 




Fig. 142. 




Fig. 143. 



46. Find the length of plank required to reach from A to B, 
as per Fig. 141. 

47. In Fig. 142 with crank pin on the top quarter the distance 
from center of driver to center of cross-head is 83% 2 in. The 
main rod is 84 in. long. Find the angle a and the stroke of the 
engine. 





Fig. 145. 



48. As in Fig. 143 find to what distance you would set dividers 
for laying off 26 equally spaced stud holes on a 233-^-in. diam. 
circle on a cylinder head. 

49. As shown in Fig. 144, what force at F is necessary to lift the 
weight of 1000 1b.? 



174 



MECHANICS AND ALLIED SUBJECTS 



50. Three holes are to be laid out as in Fig. 145. Find distances 
EC and AC. 

51. In Fig. 146, cos a = .955. Find angle a and sides x and y, 

52. The sides of a wrought-iron wedge form an angle of 2° 23'. 
What is the taper per ft. of length? 

53. Find the taper per ft. of length of a wedge the sides of 
which form an angle whose tangent is .0625. 




Fig. 146. 







Fig. 147. 



54. The shank on a drill socket has a taper of % in. in 12 in., is 
% in. in diam. at the small end and 5 in. long. What is its 
diameter at the large end (to the nearest 64th of an in.) and the 
angle between the sides? 

55. If a piece of work is to be planed to a taper of j^f ^ in. in 1 in. 
for a length of 223^^ in., how high would you block up the thin end? 



Top 
Quarieri 




Bo/fom Quarfer 



-CLofAx/e 



Fig. 148. 



56. To what distance would you set dividers for laying off 4 
equally spaced centers on an 8-in. diam. circle? 

57. In Fig. 147 showing two views of a wedge find angle a in 
degrees and minutes. How high would you block up the thin 
end for planing the tapered surface? 



MEASUREMENT OF RIGHT TRIANGLES 



175 



58. By what distance in in. is the center of the cross-head (Fig. 
148) drawn back of the middle of its stroke when the crank pin is 
on both the top and bottom quarters? 




B. 



Fig. 149. 

59. In Fig. 149, find the value of angles a and h and sides A and 

60. For Fig. 150, find the distance between the centers A and B, 



4"D/c7m. 




Fig. 150. 

(Note. — Find first the hypothenuse of each of the right triangles 
ADC and BEC.) 

61. Fig. 151 shows a cross-sectional view of a boiler shell 76 J^ 



<:5ieam 
jc^y/a-fer Line 




K H 




Fig. 151. 



Fig. 152. 



in. in diam. and the water line is 21^^ in. above the center 
line of the boiler. Find (1) the area of the steam segment, (2) 
the area of the water segment, and (3) the net water area when 
there are 465 2-in. outside diam. (0. D.) tubes. 



176 



MECHANICS AND ALLIED SUBJECTS 



62. Find the depth of a F thread in terms of the number of 
threads per in. Triangle ABC, Fig. 152, is equiangular and 
therefore equilateral. 

63. Fig. 153 represents a cross-sectional view of a boiler shell 
7 ft. 6 in. in diam. and has 482 1%-in. 0. D. tubes, the ratio 
of the height of the steam segment to the height of the water 




Fig. 153. 



segment or S to W is as 2 is to 7, thus 



W 



Find (1) the 



height and area of the steam segment; (2) the height and area of 
the water segment; (3) the net area of the water segment after 
deducting area of the tubes. 

64. Determine the distance in in. between the center of the 




McrlnRoc/-/25" 



Fig. 154. 



— Siroke 
-H -H 



55//-. 



iSfr.^ 



cross-head and the center of the stroke or point ''A," Fig. 154, 
when the crank pin occupies the six different positions marked 
1-2-3-4-5 and 6. The center line of the cylinder is 2 in. above 
the center line of the axle. How would these distances be changed, 
first by shortening the main rod, and second by lengthening the 
main rod? 



CHAPTER XVIII 



THE MEASUREMENT OF OBLIQUE TRIANGLES 



66. Methods of Working out Oblique Triangles. — In 

the following problems we are to work out triangles 
which have no right angles, that is oblique triangles. 
How we work out these oblique tri- 
angles depends on which sides and 
angles and how many of either are 
given, and which sides and angles we 
wish to find. 

In any problem we will have one of 
the four following cases. 

1. When one side and two angles are 
known, to find the remaining two sides 
and angle. 

2. When two sides and their included 
angle are known, to find the other side 
and other two angles. 

3. When only the three sides are 
known, to find all three angles. 

4. When two sides and the angle opposite only one of 
these sides is known, to find the other side and other 
two angles. 

CASE I 

Example. — Given, any one side and any two angles of 
the triangle. To find the other two sides and angle. 
In Fig. 155 suppose we liave given angle A = 75°, 
12 177 




178 MECHANICS AND ALLIED SUBJECTS 

angle B = 80"^ and side a = 30. To find angle C and 
sides b and c we proceed as follows: 

In every triangle the three angles added together 
equal 180°. Therefore angle A + angle B + angle C = 
180°. But angle A = 75° and angle B = 80°. There- 
fore 75° + 80° + angle C = 180° and angle C = 180° - 
75° - 80° or 25°. 

To find the sides b and c we have the rule. In any 
triangle any two sides have the same ratio as the sines 
of their opposite angles. Therefore the figure shows 

side b sine angle B 



also 



side a sine angle A 
side c sine angle C 



side a sine angle A 
using the last rule we have 

side c _ sine 25° side c _ .423 

~30~ ~ sine 75° ^^ 30 " ^966 

, , . . . , 30 X .423 ^_ ^ . . 

from which side c = tzw^ — or 13.14 m. 

.966 

also from the rule that 

side b _ sine angle B 
side a sine angle A 
side b _ sine 80° 

30 " sine 75° 
side b _ ^ 

30 ~ .966 

.... 30 X .985 . . _ . 

and side b = w^^ — or 30.6 in. 

.9oo 



From which 



Example.— In Fig. 156 angle A = 56° 10' angle B = 
64° 30' and side b = 21. We want to find angle C and 
sides a and c. 



MEASUREMENT OF OBLIQUE TRIANGLES 179 

Angle C = 180° - (angle A + angle B) 
= 180° - (56° 10' + 64° 300 
= 59° 20' 
side a sine A 



side b 

side a 

21 



sine B 
.830 

.903 



Therefore 



., 21 X .830 .__ 

side a = — —r^T^ — or 19.3 



.903 



To find side c, we have the rule: 

side c sin C - . side c 
thereiore 



side b sin B 



21 



.860 
.903 



, u' u 'A 21 X. 860 .__ 
from which, side c = ^^ — or 20.0 





1. From the explanation just given and using Fig. 157, complete 
the following table, drawing a separate triangle for each problem. 
The answers are given to the first two problems to help in under- 
standing them. 





Angle A 


Angle B 


Angle C 


Side a 


Side b 


Side c 


1 


60° 


65° 20' 


[54° 40'] 


18 in. 


[18.9 in.] 


[16.9in.] 


2 


80° 


30° 30' 


[69° 30'] 


[46. 6 in.] 


24 in. 


[44.1 in.] 


3 


70° 




40° 50' 






36 in. 


4 




48° 


72° 




4.2 in. 




5 


37° 


87° 15' 




48 in. 






6 




42° 16' 


85° 






52 in. 


7 




30° 


60° 


35 in. 







180 MECHANICS AND ALLIED SUBJECTS 

CASE II 

Example.-— Giveiij any two sides and their included 

angle. To find the remaining side and remaining angles. 

In Fig. 158 we have given side a = 86, side c = 72, 

and angle B = 74° 20'. In order to 
C find side b and angles A and C we 

use the following rules. 



c7^S6 




tan ^^ (angle A — angle C) _ 
tan J^i (angle A + angle C) 

side a — side c 
side a + side c 

jTiG 158 Now angle A + angle B + angle 

C = 180° 
Therefore Angle A + Angle C = 180'' - Angle B or 
180° - 74° 20' or 105° 40'. 
Then using the rule above, we have 

tan H (angle A - angle C) _ 86 - 72 
tan M(105°400 "86 + 72 

or tan >^(angle A - angle C) = tan >^(105° 40') X 
14 

158* 

The tan H(105° 40') X ^ = tan 52° 50' X ^ or 

— — r^ ■• This fraction worked out equals .117. 

If tan 3^^ (angle A — angle C) = .117 we find from the 
table of tangents that J^^ (angle A — angle C) = 6° 40' 
and angle A - angle C = 2 X [6° 40'] or 13° 20', but 
angle A + angle C = 105° 40'. Adding, 2 X angle 
A = 119° and angle A = 59° 30'. 



MEASUREMENT OF OBLIQUE TRIANGLES 181 



Angle C 



180"" — angle A — angle B or 
180° - 59° 30' - 74° 20' or 46° 10' 
To find side 6 we have as in Case I. 

side h sin B 



That is 



Therefore 



side 


a 


sin 


A 




side 


b 


sin 


74' 


'20' 


86 


sin 


59' 


\30' 


Girlo 


h 


86 X . 


963 



.862 



= 96.1 



2. From the explanation just given and using Fig. 159, complete 
the following table, drawing a separate triangle for each problem. 




Fig. 159. 

The answers in parenthesis ( ) are given for the first two problem 
to help in understanding them. 





Angle A 


Angle B 


Angle C 


Side a 


Side b 


Side c 


1 


(116° 10') 


40° 20' 


(23° 30') 


72 


(51.9) 


32 


2 


58° 40' 


(70°) 


(51° 20') 


(5.90) 


65 


5.4 


3 






70° 


120 


110 




4 


36° 40' 








98 


103 


5 




86° 




18 




24 


6 






98° 20' 


7.92 


12.3 




7 


88° 40' 








2.35 


4.25 


8 




23° 




56 




63 



182 MECHANICS AND ALLIED SUBJECTS 

CASE III 

Example. — Given, the three sides of a triangle. To find 

the three angles. In Fig. 160 we have given the sides as 

follows: side a = 4, side 6 = 6, side c = 8. To find 

angles A, B and C we use the following rules in which 

side a + side h + side c , ^ . 7 7/. ^i r 

s = ^ that IS one-nalj the sum 01 

the three sides of the triangle. 





(1) cos HA = ^/*jli?> (2) cos HE = ,/5^ 

^ _,. , ^^ , side a + side b + side c 
In Fig. 160 we have s = — — — -^ = 

^ or 9, and using rule (1) above: 

cos 3^^A = xIW^^- which equals J^^^ or .9683. 
\ o X o \ 48 

If cos 3^^A = .9683 from the table of cosines we find 

MA = 14° 28', and therefore A = 28° 56'. 

/9(9 _ 6) 

By rule. (2) above, cos l^B = \ -^, 5- which equals 

\ 4X8 

J%or .9185, and from the tables ^B = 23° 17'. There- 

fore 5 = 46° 34'. Then angle C = 180° - (angle A + 



MEASUREMENT OF OBLIQUE TRIANGLES 183 



angle B) or 180° - (28° 56' + 46° 34') which equals 
104° 30'. 

3. Using Fig. 161 and from the explanation just given complete 
the following table, drawing a separate triangle for each problem. 
The answers to the first problem are already given. 



1 

Side a 

1 


Side 6 


Side c 


Angle A 


Angle B 


Angle C 


1 


3 


5 


6 


(29° 560 


(56° 160 


(93° 480 


2 


4 


6 


5 








3 


12 


10 


9 








4 


5.6 


4.8 


3.7 








5 


5.8 


6.5 


8.4 








6 


60 


80 


48 








7 


70 


62 


39 








8 


56 


43 


36 









Put on each triangle the lengths of the sides given and the values 
of the angles found. 

CASE IV 

Example. — Given any two sides of the triangle and 
the angle opposite one of them To 
find the remaining side and other two 
angles. Suppose, as in Fig. 162, we 
have given side a = 24, side h = 30.5 
and angle A = 31° 20'. To find an- 
gle B we have by Case I: 

sin B side b . „ 30.5 X .520 ^^^^ 
or sm B = 7^-. or .6608 




sin A side a 



24 



and from the table of sines we find angle B = 41° 22'. 
But an angle and its supplement have the same sine, the 
supplement of an angle being that angle which added 
to the given angle will equal 180 degrees. For example, 




184 MECHANICS AND ALLIED SUBJECTS 

suppose we have given an angle of 38° 40', then 180° — 38° 
40' = 141° 20' and 141° 20' is the supplement of the angle 
38° 40'. 

It is possible, therefore, to have two values for angle 
B in this problem. Hence another value for angle B 
in this problem is 180° - 41° 22' or 138° 38'. Call this 
value Bi; and since we have two values for angle B, we 

must have two triangles, each having an angle A = 31° 

« 

20', a side a = 24, and a side b = 30.5, and hence two 

values for angle C and side c. Call 

these values C and Ci. Then C = 

180° -{A+B) or 180° - (31° 20' + 

^3 41° 22') which equals 107° 18'; also 

Ci = 180° - (A + ^i) or 180° - (31° 

20' + 138° 380 or 10° 2'. 

We have now to find side c and side Ci. By Case I 

we have 

side c _ sine C sideci _ sine Ci 

side a sine A side a sine A 

rpu f 'A 24 X .954 76 . . . _ 
1 here! ore side c = — - — ^^tj or 44.06 

24 X .17422 
and side Ci = ^^ or 8.04 

We will not always have two solutions, that is, two 
values for the unknown angles and sides. For instance, 
suppose as in Fig. 163 we had given side a = 40, side 
6 = 30 and angle A = 45° 30'. Then by Case I we have 

sine B side b . ^ 30 X .71325 ^^,^ 

-. — --r = -1-. or sme B = ir^ or .5349 

sine A side a 40 

and from the table of sines we have B = 32° 20'. 

The supplement of angle B is 180° - 32° 20' or 147° 
40', that is. angle Bi. But the angles A + 5i = 45° 30' 



MEASUREMENT OF OBLIQUE TRIANGLES 185 



+ 147° 40' or 193° 10' which is impossible, since the 
sum of any two angles of a triangle must be less than 
180°, hence we have only one solution to the problem: 
Angle C = 180° - {A + B) or 180° - (45° 30' + 32° 
20') or 102° 10', since 

side c sine C . , 40 X .97754 

, side c = — ~wv?ir^ — or 54.8 



side a sine A 



.71325 



Again, suppose as shown in Fig. 104 we had given 
side a = 32, side b = 44, and angle 5 = 112° 30'. 

^, , ^ -r , sine A side a 

Ihen by Case 1 we have ^' = -r-, — r- 

sme B side o 

Now sine 112° 30' = sine (180° - 

112° 30') or sine 67° 30' (since the 

sine of any angle less than 180° 

equals the sine of 180° minus that an- 

, ^ ^, . . ; 32 X .92388 
gle). 



Therefore sine A = 



44 



or .67191, and from the table A = 

42° 13'. Angle C = 180° - {A+ B) A 

or 180° - (42° 13' + 112° 30') or 25° 

^-, , . side c sine C 
17 ; also since 




Fig. 164. 



side a sine A 



then 



. , 32 X .42709 _^ _ 

side c = w>^vK^ .or 20.34. 

.d7191 

It is evident, since the given angle is greater than 90°, 
that there is only one solution to this problem. 

We now have the following rules by which we may tell 
at a glance how many solutions there are to a problem 
under this case (Case IV). 

1. If the side opposite the given angle is less than the 
other given side there are two solutions. 

2. If the side opposite the given angle is greater than 
the other given side there is only one solution. 



186 



MECHANICS AND ALLIED SUBJECTS 



3. If the given angle is greater than 90° there is only 
one solution. 

4. From the explanation just given and using Fig. 165 as a 
guide complete the following table, drawing a separate triangle for 
each problem. The answers to the first problem are already- 
worked out. 






Angle A 


Angle B 


Angle C 


Side a 


Side b 


Side c 


1 


32° 10' 


5 = 45n3' 
Bi = 134°47' 


C = 102°37' 
Ci = 13°3' 


6 


8 


C = 11 

ci = 2.54 


2 






26° 40' 




10 


7 


3 




102° 50' 


' 




73 


65 


4 






39° 10' 


28 




33 


5 




59° 




16 


14 




6 




120° 50' 






48 


42 


7 






83° 20' 


47 




51 


8 


128° 30' 






38 




30 



67. Applications of Triangular Functions to Oblique 
Triangles. — Following are applications to the solution 
of practical problems of the preceding rules for oblique 
triangles. 

6. In Fig. 166 is shown the center lines of a roof truss. Find 
the angle C and the length of the rafters a and b. 

6. In Fig. 167 find the radii Ri and R2 for laying out the three 
centers A, B, and C so that they may be located as shown. 



MEASUREMENT OF OBLIQUE TRIANGLES 187 

7. In Fig. 168 the diameter of a boiler is given as 76^4 in. 
and angle ''A'' is 150°. Find the length of water line ''x. " 

8. It is desired to find the distance between two points A and B 
on opposite sides of a river. If the distance from 5 to a point C, 
Fig. 169, is found to be 128 ft. and the angles ABC and BCA are 





Fig. 166. 



Fig. 167. 



found to be 43° 40' and 50° 20' respectively, what is the distance 
ABl 

9. It is desired to find the height of the stack shown in Fig. 
170. The angle of elevation of a point in the horizontal plane 





Fig. 168. 



Fig. 169. 



of the base of the stack is 54° 30', and the angle of elevation at a 
point 50 ft. farther away is 44° 20'. Find the height of the stack. 
10. The angle of elevation of the top of a stack at a point A, 
as shown in Fig. 171, is 69° 20' and the angle of elevation at a point 
B, 50 ft. farther away, is 54° 50'. If the angle of elevation at 



188 



MECHANICS AND ALLIED SUBJECTS 



point A of the top of the building is 45° 20', find height of stack 
and height of building. 

11. Two trains start from the same station at the same time 
traveling on tracks which intersect at an angle of 64° 20'. If the 





SO 



Fig. 170. 



'^''70' 



Fig. 171. 



trains travel at the rate of 35 and 50 m.p.h., respectively, how 
far apart are they at the end of 45 min. ? 

12. A Ji-in. steel plate has the form of a parallelogram as shown 
in Fig. 172. If the sides are 64 in. and 40 in. in length and the 



et s 




Fig. 172. 




3^ 3 3; 



Fig. 173. 



longer diagonal is 88 in. in length, find the angles A, B, C, and Z), 
the area of the plate, its volume, and also its weight if 1 cu. in. of 
the metal weighs .283 lb. 

13. In. a steam engine the crank OA (Fig. 173) is 13 in. long, and 



MEASUREMENT OF OBLIQUE TRIANGLES 189 

the connecting rod AB is 125 in. long. How far has the center B of 
the cross-head moved from its head end dead center position Bi 
when the crank pin is midway between C and Dl 

14. Given a 3^-in. steel plate with the shape as shown in 
Fig. 174. Find its area in sq. in. and its weight if steel weighs 
.28 lb. per cu. in. 



Ar^ 



29' 20' 24!'30'B 




Fig. 174. Fig. 175. 

15. Given a %-in. steel plate as shown in Fig. 175 with its long- 
est side 80 in. and width 40 in. Find the non-parallel sides, its 
area in sq. in., and its weight. 



CHAPTER XIX 
ELECTRICITY 

68. Definitions and Calculation of Resistance. — In 

making electrical calculations we need to understand the 
meaning of electrical '' resistance^ ^ and how to calculate 
it. Every wire or electric circuit or portion of a circuit 
which conducts an electric current offers a ^Wesistance^^ 
to the flow of the current through it. We may consider 
this resistance in a way like mechanical friction. Less 
water will flow in a pipe that is rough than in a smooth 
pipe because friction cuts down the flow. Less electric 
current will flow in a wire or ^^conductor, ^' as it is called, 
having a high resistance than in a low-resistance con- 
ductor. The effect of the electrical resistance is to cause 
a part of the electrical energy to be changed into heat. 
Every electrical conductor is therefore heated by an 
electric current flowing through it. The less the re- 
sistance of a conductor the less the heat produced by a 
given current. Electrical resistance is measured in 
ohms. 

The resistance of a conductor depends on its material, 
that is, whether copper, aluminum or iron, upon its length, 
area of cross section and its temperature. Copper is a 
better conductor of electricity than either aluminum or 
iron, that is, it offers less resistance to the flow of the 
electric current through it. If copper wire is heated its 
resistance increases. The resistance of a wire increases 
as its length is increased and decreases as its area of 

190 



ELECTRICITY 191 

cross section is increased. In electric wiring it is neces- 
sary to have wire large enough in cross section to carry 
the current required without overheating. If a wire too 
small is used there is danger from fire and insurance rules 
limit the size of wire for different kinds of work. In 
calculating the resistance of wires of circular cross sec- 
tion, their area is often found in circular milSj by squaring 
the diameter expressed in thousandths of an inch. Thus 
a wire 3^^ in. or .25 in. in diam. has (.25 X 1000)^ 
or 62,500 circular mils in it. This is not the area as 
we determine the area of a circle, but is a convenient 
way of expressing relative areas of wire cables, etc. To 
find the resistance of copper wire at ordinary tempera- 
ture we use the following rule : 

10.8 X length of wire in ft. 



Resis. in ohms = 



area of cross section in cir. mils 



Using this rule the resistance in ohms of 1 mile (5280 ft.) 
of copper wire .12 in. in diam. is equal to 

10.8 X 5280 10.8 X 5280 _ ^ q. , 

(.12 X 1000)2 ^^ 120 X 120 ~ ^'^^ ^^^^ 

For another example, the resistance of 2000 ft. of copper 
wire .20 in. in diam. is 

10.8X2000 10.8X2000 ,, , 

or — 777-7^7;7^ or .54 ohm. 



(.20X1000)2" 40,000 

Ten miles of a 100,000 circular mil copper cable has a 

resistance of 

10.8 X 10 X 5280 , ^^ ^ 
foo;ooo ^' ^'^^ ^^^' 

69. Resistances in Series. — If a number of electrical 
conductors are arranged one after another in a circuit as 
in Fig 176 so that the same current flows through all of 



192 MECHANICS AND ALLIED SUBJECTS 

them, they are said to be ^^in series'^ and their total 
resistance is equal to the sum of their separate resistances: 

Total resistance = resistance of A + resis. of B 

+ resis. of C 

A 3 C 
yW vAA/^ vAA/^ 

Fig. 176. 

70. Resistances in Parallel. — If the conductors are 
arranged side by side as in Fig. 177 so that the current 
divides in going through them and each conductor takes 
only a part of the total current, tkey are said to be ^^in 
paraller^ and the reciprocal of the total resistance is 
equal to the sum of the reciprocals of the separate 
resistances : 



Total resistance resis. A resis. B resis. C 

A 



AAW 



^AAAV 



Fig. 177. 

For example, if a copper cable of .10 ohm resistance is 
in parallel with an aluminum cable of .85 ohm resistance 
the combined resistance is found as follows : 



total resistance .10 .85 

or r-T-T— ".-I = 10 + 1.18 or 11.18 

total resistance 

and total resistance = tyTq ^^ .0895 ohm. 



ELECTRICITY 193 

For transmitting power, copper, aluminum, steel, and 
phosphor bronze wire are used, copper and aluminum 
principally. For telephone and telegraph work steel 
and iron wires are used. For protective devices such 
as fuses and for coils of electrical measuring instruments, 
various alloys or mixtures of metals are used. 

Copper is used to a large extent as an electrical con- 
ductor on account of its comparatively low resistance. 
Aluminum has an advantage over copper in its lighter 
weight for the same conducting power. This in trans- 
mission lines reduces the weight on insulators, poles, 
and cross-arms. 

PROBLEMS ON RESISTANCE 

1. Find the resistance of 5 miles of transmission line with con- 
ductors .187 in. in diam. 

2. It is desired to extend the above line 2 miles with conductors 
.125 in. in diam. What is the resistance of the line added and o^ 
the total line? 

3. Find the resistance of 12 miles of 150,000 circular mil copper 
cable. 

4. An aluminum cable of 12 ohms resistance is put ^^in parallel'^ 
with the copper cable of Prob. 3. What is the resistance of the 
two in parallel? 

5. A copper wire has a resistance of 20 ohms. Another wire 
of the same material is twice as long as the first and twice as large 
in cross section. How does its resistance compare with that of 
the first? 

6. Give as many examples as you can of the use made of the 
heating effect of an electric current flowing through a resistance. 

7. The resistance of a line wire is 4 ohms and a second wire of 
the same size is run along and ^'tied in^' with the first, what is 
the resistance of the two taken together? 

8. The resistance of a copper wire is 5 ohms. What is the re- 
sistance of another wire of the same material three times as long 
and of one-third the cross-sectional area of the first? 

13 



194 



MECHANICS AND ALLIED SUBJECTS 



9. What metals are used as electrical conductors besides copper 
and aluminum? Explain in what kind of service they are used. 

10. From the table on this page what number wire would you 
use for concealed work to carry a current of 22 amperes? What 
part of an inch is the diameter of this wire, and what does it 
weigh per mile if insulated ? 



71. 


Safe Carrying Capacity of Wires.- 


— 




Gauge 

No., 

B. & S 


Diameter, 
mils. 1 

1 


Ohms per 
1000 ft. 


No. 

amperes, 

open work 


No. 

amperes, 

concealed 

work 


Lb. per 
1000 ft., 
bare 


Lb. per 

1000 ft., 

insulated 


18 


40 


6.3880 


5 


3 


4.92 


18 


17 


45 


5.0660 


6 


4 


6.20 


21 


16 


51 


4.0176 


8 


6 


7.82 


25 


15 


57 


3.1860 


10 


8 


9.86 


31 


14 


64 


2.5266 


16 


12 


12.44 


38 


13 


72 


2.0037 


19 


14 


15.68 


43 


12 


81 


1.5890 


23 


17 


19.77 


48 


11 


91 


1.2602 


27 


21 


24.93 


64 


10 


102 


.99948 


32 


25 


31.44 


80 


9 


114 


. 79242 


39 


29 


39.65 


97 


8 


128 


.62849 


46 


33 


49.99 


116 


7 


144 


.49845 


56 


39 


63.03 


118 


6 


162 


.39528 


65 


45 


79.49 


166 


5 


182 


.31346 


77 


53 


100.23 


196 


4 


204 


.24858 


92 


63 


126.40 


228 


3 


229 


. 19714 


110 


75 


159.38 


265 


2 


258 


. 15633 


131 


8S 


200.98 


296 


1 


289 


. 12398 


156 


105 


253.43 


329 





325 


.09827 


185 


125 


319.74 


421 


00 


365 


.07797 


220 


150 


402 . 97 


528 


000 


410 


.06134 


262 


181 


508 . 12 


643 


0000 


460 


.04904 


312 


218 


640.73 


815 



72. Ohm's Law, Calculation of Current, Voltage and 
Resistance. — In order to force an electric current through 



ELECTRICITY 195 

a wire or conductor in spite of its resistance it is necessary 
to have an electric pressure just as it is necessary to have 
water pressure or ^^head'^ to force water through a pipe 
in spite of friction. 

Electric pressure is measured in volts and is indicated 
by an instrument called a voltmeter. Instead of electric 
pressure we sometimes speak of the 
'^ voltage ^^ of a circuit or the ^^elec- \\\\- 

tromotive force ^^ of the circuit. The 
three terms ^^ electric pressure/' '^vol- Ammeier 

tage/' and ^^electromotive force'' all 

mean the same pressure required to lOVolis 

force an electric current through a cir- Je^rmmals 

cuit. The amount of current which 

will flow in a circuit of given resistance Fig. 178. 

and under a given pressure is found 

from the following rule known as Ohm's Law (named 

from Dr. Ohm): 

voltage 




Current = 



resistance 



If the voltage is in '^ volts" and the resistance in 
^^ohms" the current is in ^^ amperes." For example, 
if, as in Fig. 178, a 20-volt battery is connected to a 

wire of 5 ohms resistance a current of -^ or 4 amperes 

will flow through the circuit, and may be read on the 
^^ current meter" or '^ ammeter." 

For example, if an incandescent carbon filament lamp 
having a resistance of 220 ohms is put on a 110- volt 

circuit, 'the current which it will take equals ™^ or .5 

ampere. From the rule that 



196 



MECHANICS AND ALLIED SUBJECTS 



(1) Current = 



voltage 



it follows that 



resistance 

(2) Voltage = current X resistance, and also 

(3) Resistance = r* 

current 

Rule (2) enables us to find the voltage across a known 
resistance through which a known current flows, and rule 
(3) enables us to find the resistance of a circuit with a 
known current flowing under a known pressure. 

In Fig. 179, three lamps are arranged ^^in series'' 
across 110-volt mains. Here the current through each 



IfO Volh 



110, Ohm Lamps •. 



■^ 0.33 Ampere 



r 



Fig. 179. 



3^1 



- llOVol-fa — H 
la 



la 



la 



O 



Lamps 



o 



o 



Fig. 180. 



\3a 



Zamp is the same and if each lamp has a resistance of 

110 ohms, the three ^^in series'' have a resistance of 

3 X 110 or 330 ohms. The current through the lamps 

.1 f 1 the voltage 110 ^^ 

therefore equals -r ^7-^ — ^ or ^^7^ or .33 ampere. 

the resistance ooO 

From rule (2) the voltage across each lamp equals 

current X resistance, or .33 X 110 or 36.7 volts. 

Fig. 180 shows three 110-ohm lamps arranged ^^in 

parallel across 110-volt mains. The current through 

the voltage 110 

or 1 ampere. 



each lamp equals . ^ ^ , — -, . r. 

resistance 01 lamp 110 

The three lamps together therefore take three amperes, 



ELECTRICITY 197 

The arrows indicate the direction of flow of the current 
and the figures, the number of amperes in the different 
parts of the circuit. 

PROBLEMS 

11. What is the resistance of a magnet winding through which 
1.5 amperes of current flows under a pressure of 110 volts? Draw 
a figure showing the magnet in circuit? 

12. If a 32-c.p. carbon lamp of 110 ohms resistance is connected 
across a 110-volt circuit, what current flows through it? How 
much current do ten of these lamps take when arranged '*in par- 
allel" on a 110-volt circuit? 

13. How much current would each of ten 220-ohm lamps take 
if put ''in series" in a 550-volt circuit? 

14. A carbon filament lamp operating on 110 volts takes .95 
ampere when first put in circuit and .93 ampere after burning 
for 10 min. What is the change in resistance due to heating the 
filament? 

16. Three 110-ohm lamps are connected in series and these in 
series with a parallel group of five 220-ohm resistances. What is 
the resistance of the total number of lamps as arranged? 

16. In Prob. 15 what is the voltage across each part of tKe 
circuit when 110 volts are impressed on the total circuit described? 

17. Two thousand feet of No. 18 B. & S. gauge copper wire is 
to be wound on a magnet and operated on a 110-volt circuit. 
What resistance must be put ^'in series" with this winding in 
order that the current may be kept within safe limits? (See 
table, page 194.) 

18. What is the weight of 10 miles of No. 00 B. & S. gauge 
trolley wire? 

19. What is the total current taken by ten lamps in parallel if 
each has a resistance of 121 ohms and operates on a 110-volt circuit? 

73. Calculation of Power and Energy in an Electric 
Circuit. — The power consumed in a direct current circuit 
and the power taken by an incandescent lamp, whether 
on a direct or alternating current circuit equals the product 



198 



MECHANICS AND ALLIED SUBJECTS 



of the current flowing through the circuit and the voltage 
across the circuit, that is, 

power = current X voltage 

When the current is in '^amperes'' and the voltage in 
''volts,'' the power is in watts. 

For example, if an incandescent lamp, as in Fig. 
181, across 110-volt mains and has 1 ampere flowing 



//O Volis 
'Ammeter 




3a^ 


< 1 10 Vol f 5 — > 


\5a 


2a^ 




\2a 


.la\ 




Ma 




Fig. 182. 





through it, the power in watts equals the voltage 110 
multiplied by the current 1 or watts = 110 X 1 or 110. 
In Fig. 182, three 110-ohm lamps are shown in par- 
allel 2^0x0^^ 110-volt mains. By the rule for finding the 
current, we have the current taken by each lamp = 



the voltage across lamp 
resistance of the lamp 



or 



110 
110 



1 ampere. 



The power taken by each lamp = current through lamp 
X voltage across lamp = 1 X 110 or 110 watts. The 
three lamps together therefore take 110 X 3 or 330 
watts. The arrows and numbers in the figure show the 
direction of flow of the current and the number of 
amperes in each part of the circuit. 

One thousand watts is called a kilowatt. In the 



ELECTRICITY 199 

330 
example above 330 watts or fT^ = .330 of a kw. is the 

power consumed constantly. 

The energy consumed in an electric circuit equals the 
product of the power and the time during which the 
power is taken. If the power is in watts and the time 
in hours, their product gives the energy in watt-hours. 
If the power is in kilowatts and the time in hours their 
product gives the energy in kilowatt-hours which is the 
unit used by electric light and power companies. If 
lights which take .30 kw. are kept burning for 3 hr. the 
energy they consume in this time equals kilowatts X 
hours or .30 X 3 or .90 of a kw.-hr. 

Electrical energy is sold at prices ranging from a frac- 
tion of a cent to 18 or 20 cts. per kw.-hr. (kilowatt- 
hour). At 8 cts. per kw.-hr. the cost of the energy 
consumed by the lamps taking .90 kw.-hr. is .90 X 8 or 
7.2 cts. 

PROBLEMS 

20. What power is taken by a 110-volt carbon filament lamp 
with a current of .98 ampere? 

21. A tungsten lamp takes .545 ampere at 110 volts. What is 
its wattage? 

22. What is the difference in power taken by a group of five 
110-volt carbon filament lamps each taking 1 ampere and a cluster 
of three 110-volt tungsten lamps each taking 60 watts? 

23. Calculate the kilowatt-hours of energy consumed in 5 
hr. by three 100-watt tungsten lamps. 

24. How much energy is taken in 2 hr. by twelve 60-watt 
lamps? 

25. What is the cost per month of 30 days of 2 hr. of lighting per 
day, for an entire building with energy of 8 cts. per kw.-hr., if 
there are twelve 100-watt lamps and the equivalent of twenty 40- 
watt lamps in the building? 

26. Find the cost per month (30 days) for house lighting, with 



200 MECHANICS AND ALLIED SUBJECTS 

4 carbon filament lamps each taking 1 ampere at 110 volts and 
burning for an average of 5 hr. per day. Take the cost of energy 
at 8 cts. per kw.-hr. 

27. What kinds of incandescent lamps are in general use? 
State the advantages and disadvantages of each. 

28. An office is first lighted with six 16-c.p. carbon filament 
incandescent lamps taking 3.1 watts per candle. The carbon 
lamps are later replaced by four 60-watt tungsten lamps. What 
is the saving per month of 30 days with the lamps burning 5 hr. 
per da}^ if energy costs 8 cts. per kw.-hr.? 

29. How does the first cost of carbon and tungsten lamps com- 
pare? Is there a saving in using tungsten rather than carbon 
lamps? 

74. Electrical Horsepower. — Horsepower means the 
rate at which work is done. For example, 33,000 ft.- 
Ib. of work done per min. or 550 ft. -lb. per sec. repre- 
sents one horsepower. In electrical terms one horsepower 
(abbreviated h.p. or HP) equals 746 watts. To find 
the horsepower taken by a direct-current motor multiply 
the voltage on the motor by the current taken by the 
motor and divide the product by 746, that is 

, „ ^ Volts on motor X current intake 
horsepower oi a motor = ^t^ 

Example. — Find the horsepower consumed by a 110- 
volt motor taking a total current of 10.3 amperes. 

^ 110 X 10.3 ^ _ . 
Horsepower = jZr ^^ Ans, 

To find the horsepower delivered by an electric generator , 
multiply the voltage at the generator terminals by the 
current output and divide the product by 746, that is 

, ,. , Volts at gen. ter. X curr. out. 
horsepower oi a generator = ^40 ~~~~~~~ 



ELECTRICITY 201 

Example. — An electric generator with 110 volts at its 

terminals supplies 25 amperes to a lamp board. What 

horsepower is it giving out as useful power? 

„ 110 X 25 _ .. . 

Horsepower = — ^^ — or 3.69 Arts. 

To find the horsepower -hours, multiply the load in horse- 
power by the time in hours during which the power is 
supplied. 

Example. — A generator supplies 13 amperes at 120 
volts for 5 hr. what is its horsepower hour output? 

Horsepower hours = — wj^ — X 5 = 10.45 Arts. 

PROBLEMS 

30. What horsepower generator is necessary to supply 30 
amperes at a pressure of 110 volts? 

31. A motor driving a lathe takes 6.4 amperes at 220 volts, 
what is the horsepower supplied to the motor? 

32. Three direct-current and voltage ratings are as follows: 
10 amperes, 110 volts; 15 amperes, 220 volts; 30.5 amperes at 
110 volts. Find the corresponding horsepower rating of each 
of these motors when working at full load. 

33. Five tungsten lamps, each take .95 ampere at 110 volts, 
what is the number of watts and horsepower taken by the lamps? 
If the lamps burn for 5 hr. what energy do they take in watt-hours, 
kilowatt-hours, and in horsepower-hours? 

34. What kind of instruments are used for measuring an electric 
current? Find out over what range these instruments are made 
to read. 

35. What kind of instruments are used for measuring electric 
pressure or voltage? Find out what range these instruments 
have. 

36. How are electrical ''power" and "energy" measured? 

37. Find out what different kinds of electric lamps are in general 
use, with the advantages and disadvantages of each. 

38. Wliat rules do you know in regard to personal safety and 
protection of machines when working about electrical circuits? 



202 MECHANICS AND ALLIED SUBJECTS 

75. Methods of Charging for Electrical Energy. — • 
(a) Flat Rate System. — In the flat rate system the 
customer is charged a certain amount per month for each 
lamp installed. No meters are used to determine the 
amount of energy consumed. With this system a waste- 
ful consumer pays no more for the same number of lights 
installed than an economical user. This system saves 
the cost of meters, their repair, and simplifies the com- 
pany's bookkeeping. A scale of discounts is usually 
arranged increasing with the number of lamps installed. 

(6) Meter System. — In this system the consumer is 
charged for the energy which he uses, that is, for the 
number of kilowatt-hours which he consumes in a certain 
time. This method would seem to be a just way of 
charging for energy, but the time of day that power is 
taken is of as much importance as the amount taken. 
Power taken at a time when the station load is heavy is 
more expensive for the company than power taken at a 
time of light load. The cost of power depends on the size 
of the power station, the amount of power taken, and the 
time at which it is taken. In view of these facts a ^Hwo 
rate'' meter is sometimes used with two sets of dials, one 
reading at times of light station load and the other at 
times of heavy station load. The consumer is then 
charged one rate for the power taken when the station is 
working light and a higher rate for power taken at times 
of heavy load on the station. 

A ^^ maximum rate" meter is also used in some cases. 
This meter gives the greatest load or power which the 
consumer has taken during a certain period. The charge 
is then made on a basis of the greatest amount of power 
taken at any one time as well as on the total number of 
kilowatt-hours consumed. 



ELECTRICITY 203 

A certain^ charge is usually made whether the customer 
uses any power or not. This is because the customer has 
power at hand any time he wishes to use it, and a charge 
is made for this service. 

PROBLEMS 

39. Find the total resistance of the circuit shown in Fig. 183. 

40. If the circuit of Prob. 39 is connected across 110-volt mains, 
what is the current through the circuit and the voltage across 
each portion of the circuit? 



4' I/O Ohm Lamp:. 
X X X X 



:^ 3-20 Ohm 



^ 



^ 



X ■ X X 

Hesi stances 



3-220 Ohm Lamps 
Fig. 183. 

41. Find the difference in power taken by eight 220 ohm lamps 
operating on a 220-volt circuit and sixteen 100- watt lamps. 

42. If a current of 8.2 amperes flows through a resistance on 
110 volts, how much more resistance must be put in circuit to 
reduce the current to 4.5 amperes? 

43. How many horsepower-hours are equivalent to 2000 kw.-hr. ? 
How many kilowatt-hours are consumed by a lamp load of 125 
amperes at 220 volts in circuit for 4 hr. ? 

44. Calculate the power in watts taken by thirty 110-volt carbon 
lamps connected in parallel if each lamp takes 1.5 amperes. 

45. Calculate the power taken by 20 lamps in series on a 1100- 
volt circuit if 15 amperes flow through the circuit. 

46. Calculate the power in kilowatts taken by ten 110-volt 
lamps in parallel across a 110-volt circuit if each takes .9 ampere. 

47. Ten lamps taking 1 ampere each are connected in parallel 
across 110-volt mains. What is the resistance of each lamp and 
the power taken by the circuit? 

48. Four 110-volt carbon lamps taking 1 ampere each were 
replaced by four 40-watt tungsten lamps. If the cost of energy 
was 8 cts. per kw.-hr., how much did they gain or lose per hour by 
the change? 



204 



MECHANICS AND ALLIED SUBJECTS 



49. Find the difference in kilowatt-hours taken in 5 hr. by ten 
220-ohm lamps on 110- volts and eight 60-watt tungstens. 

50. Find cost of operating the circuit shown in Fig. 184 per 
month of 30 days and 4 hr. per day, energy 10 cts. per kw.-hr. 

51. What system of charging for electricity do you consider 
the best, and why? 



■/^OK -H 6- 40 Waii Tungstens 



8-100 WaH- Carbon Larnps 



H U M U" 



Fig. 184. 



76. Comparison of Water and Electric Systems. — A 

very close and interesting comparison can be made be- 
tween a water system with its piping, gauges, pumps, 
motors and so forth and an electric system with its wir- 
ing, meters, generators, motors and so forth. In Figs. 
186 and 187 diagrams are shown of typical water and 
electric systems, with the corresponding parts in the two 
systems marked and explained as follows: 



ELECTRICITY 



205 





Mt Id 


I'i i 

> u 


ft 


K } __-^-^pp 


^^MHgi^Hjfli*'^ 


J I V- 




t« \ 


1 1 ^S 




Ml 
"'J § 

1^ 


<^ '~™-" „ 








9 


^^■l ^ 




f ' m^ 


"i li 

1 ^ 


^^■^^H 


1 — -^^j 




^B ^ ...... m L / 


■H 





O 



c3 



CI 

O 
m 



:3 



00 



t-l 



206 



MECHANICS AND ALLIED SUBJECTS 




Eleciric 
Mofor 



Cenirifuga/ 
Pump 



^t # 



Waier Ex^hausi 
Supple/ 



Fig. 186. 



(Eleciric Mofors 

Q Volimefer \p 
YX''5mich^^ 



Eleciric Mohr 

\\)Voli' 
meier 

* Elec. Oeneraior 



Waii' 
meier 



*t 



ijijijip 



|i|i|i 

Siorage 
BaOerg 



Fig. 187. 



ELECTRICITY 



207 



Water system 


Corresponding part in electric system 


Electric motor for driving cen- 
trifugal pump. 


Electric motor for driving elec- 
tric generator. 


Centrifugal pump. 


Electric generator. 


Pressure gauge, giving water 
pressure. 


Voltmeter giving electric pres- 
sure. 


Gate valve for shutting off water . 


Switch for opening electric 
circuit. 


Water meter for measuring quan- 
tity of flow of water. 


Wattmeter for measuring the 
power taken by the circuit. 


Globe valve for shutting off water 
supply to motor. 


Switch for cutting off electric 
current to electric motor. 


Water motor for doing mechan- 
ical work. 


Electric motor for doing me- 
chanical work. 


Storage tank for water pumped 
by centrifugal pump. 


Storage battery for electricity 
supphed by generator. 



CHAPTER XX 
STRENGTH OF MATERIALS 

77. Definition. — All materials have a property or 
characteristic which we call ^^ strength/^ which is quite 
different for different materials such for example as wood 
and steel. In order to know what kind of material to 
use for a particular piece of work and how much of it 
to use, that is how large a piece is required in order that 
the completed work or job shall be sufficiently strong to 
serve its purpose, it is necessary to know what the 
strength of the various materials is and how this strength 
differs for the different methods in which pressure may 
be brought on the material. 

The average person is likely to speak only relatively 
regarding the strength of materials, namely that one 
material is so many times stronger than another. It 
is necessary, however, in working with machines and in 
general construction work to have more definite knowl- 
edge of the strength of materials. This has led to the 
construction of testing machines, capable of measuring 
directly in pounds the strength of the materials under 
various conditions. These machines are described later 
on in this chapter. 

The object of this chapter is to convey some idea of 
the strength of materials, from results secured by actual 
tests and how these results are used in calculating the 
size and amount of material required. 

208 



STRENGTH OF MATERIALS 209 

This chapter also brings out some of the effects of 
forces appUed to materials in changing their size and 
shape, and also in merely producing pressure in them 
without changing their form. 

78. Stress and Strain. — Whenever a pressure or a load 
is put upon any piece of material it tends to change the 
shape of the piece. The material itself resists this 
change of shape and in so doing exerts a force opposite 
to the load or pressure applied to the material. For 
example, if we take a piece of string and, holding it in 
our hands, pull on it with an increasing force the string 
will finally break, but not until it has straightened out 
and become taut. After this there is a gradual stretch 
until the fibers pull apart. Our effort or pull on the 
string has been opposed by a resistance in the fibers of 
the string. This resistance is called stress. Stress may 
therefore be defined as the resistance which a material 
offers to the action of an external force, pressure or load 
which tends to change the shape of the material. 

If a 2-ton weight is being lifted by a rope the stress 
produced in the rope is 4000 lb. 

The stretching of the string lengthened or elongated it 
and produced a change of its form. This change of form 
in the fibers of a material is called strain. 

In construction work the materials used must be of 
sufficient size and strength so that the loads applied 
are not sufficient to change their shape, otherwise the 
parts are likely to break or '^rupture'' and disastrous 
results follow. 

79. Kinds of Stresses. — There are three general kinds 
of stresses, namely (1) Tensile stress (or puUing stress), 

(2) Compressive stress (pushing or crushing stress), 

(3) Shearing stress (cutting stress) . 

14 



210 



MECHANICS AND ALLIED SUBJECTS 



There are two other kinds of stresses which in reahty 
are combinations of those given above. These are (a) 
Torsion or twisting which is a kind of shearing stress 



^Threads 
in Shear 




Plaiem 
Tension 



EYE BOLT 



STAY BOLT 




Rivets 
in 5 ilea r 



BOILER SHELL 



Fig. 188. 



and (6) Bending or flexure which is a combination of 
tension and compression with or without shearing. The 
examples (Figs. 188, 189 and 190) show the various stresses 
of materials in service. 



rfi M 




Plaie Sliear 



Puncli 
Fig. 189. 



bhear'' 
Siiear on Pin 



Since the tensile stress set up in a material is due to a 
pull, the tensile strength of the material is a measure of 
the resistance offered by its fibers to being pulled apart. 



STRENGTH OF MATERIALS 



211 



Since the compressive stress is a stress set up in a 
material due to a push, thrust, or crushing force, the 
compressive strength is a measure of the resistance offered 
by the fibers of the material to being crushed. 

A shearing stress is one set up in a material due to a 
force acting at right angles to the material and the 
shearing strength is a measure of the resistance of the 
fibers of the material to being cut or sheared off. 



Pulley 
•QwfsHng 



^.. Shaff 



Beli- 



P/sfo/7 rod in al-fernafe 
compression and 
'^ iension 



Bending. ■ 




( Cylinder in i^ension 
' same as boi/er shell 



777777777i 



' Pw// 



vv 



Fig. 190. 



80. Ultimate Strength. — If samples of different ma- 
terials are loaded until they break or rupture we can find 
out how much each material will stand. The ultimate 
strength of a material is the number of pounds per square 
inch of cross section of the material at which it breaks. 
This value varies between wide limits for different 
materials and often varies considerably for the same 
material. 

The ultimate strength of a given material is different 
for the different stresses of tension, compression and 
shear. 



212 



MECHANICS AND ALLIED SUBJECTS 



81. Testing Machines. — For testing materials to find 
their ultimate strengths, testing machines as shown in 
the following figures are used. 

Fig 191 shows a machine used for tension and com- 



1 








h 


1- 


SSHISfl^^^w 


m^^ 


1 


I 


1 


1 


|Wi^ 


1 


1 


1 



Fig. 191. 



pression tests. The figure shows a piece being tested 
in tension. 

The test specimen is placed in the machine as shown 
between heads (H) and (C) and is held firmly in place 
by means of wedge grips. Head (C) is fixed to the body 



STRENGTH OF MATERIALS 



213 



of the machine (A) by frame (D). The bed (A) rests on 
a set of levers. Through the head (H), two long screws 
are threaded which are geared to bevel gears shown in- 
side of the case (J5). A belt connected to a pulley which 
is fastened on a shaft to which the gears are fastened, 
drives the machine, which causes the screws to revolve 
slowly. This action moves (H) down causing a pull on 
the test piece and hence on the head (C), the pull being 
transmitted to (A). 

(A) resting on lever (Z>) forces (D) down and by the 
system of levers, causes the lever on which (F) rests to 
move up. The system of levers is kept balanced by 



■t; 



D^ 



Fixed Fulcrum, 
d 



# 






Loadi 



B^ 



^ 



<- o ^J<- 



-H 



/yxec/ Fu/crum 

Fig. 192. 



moving the weight {F) out along the graduated bar. 
{E) and (ilf ) are weights used for balancing and for ad- 
justment. All readings of the load applied and also the 
breaking load, are registered by means of (jP), the lever 
on which it moves being marked off in pounds so as to 
read directly the pull exerted on the test piece. 

When a piece is to be tested in compression, it is placed 
between the head {U) and (A) and it can be seen that as 
the head (//") moves down, the piece is compressed and 
the load registered as before. 



214 



MECHANICS AND ALLIED SUBJECTS 



The system of levers is shown in Fig. 192 giving an 
outHne of their arrangement. 

As the load is applied to the test specimen, it pushes 
down on (A) and tends to move lever (BC) around the 
fixed point at (5). The force at (C) then pulls down 
on the lever (DE) which is pivoted at (E). The lever 
(FG) is connected to the lever (DE) by the link (FD), 




Fig. 193. 



and as the force at (D) pulls down on the end (F) of 
lever (FG), the end (G) tends to move up. The end (G) 
is kept balanced between (Y) and (F) by moving the 
weight (F) out along the lever. The object of the 
system of levers is merely to reduce the large load act- 
ing at (A) to a small load acting at (F). Nearly all 
testing machines are built on the principle of a system 



STRENGTH OF MATERIALS 



215 



of levers and the methods are worked out in a manner 
practically similar to that just described. 

For bending tests of wood specimens a machine similar 
to that shown in Fig. 193 is used. The lever arrange- 
ment is similar to that of the tension machine. The 
method of supporting the specimen and recording the 
load on the beam and the deflection of the beam can 
be readily seen. This machine is operated by hand. 




Fig. 194. 



Pieces to be tested in torsion are placed in a machine 
similar to that shown in Fig. 194. The specimen is 
fastened in grips (A) and (B). (B) is fastened to {M) 
which has an arm (C) attached to it. By means of the 
belt shown at the left and the cone pulley and system 
of gears, (A) can be rotated. As it moves it tends to 
turn {B) and (M) and (C) tends to rise. Arm (C) then 



216 



MECHANICS AND ALLIED SUBJECTS 



pulls up on (D) which is a lever pivoted to the frame 
and so that the force is carried to the lever on which 
(H) rides. The lever is kept balanced by moving weight 
(H). The weight (F) serves as a balance. The system 
of levers here shown is practically the same as for the 
machine shown in Fig. 193 and the method of recording 
the load is similar. 

82. Standard Test Pieces.— Figs. 195, 196 and 197 
show a few of the standard test pieces or specimens used 
in finding the strength of materials. 

The specimen used should in any case be as near an 
average specimen of the material as possible. If a 



yr 



j7 



►<- 



K 



TV 



Sc/ 



4d 



C(/lindrical 



r\ 



4b 



' Aboui- lO^area 

or lOibci Rectangular 

Fig. 195. 



•>\d'& 



shipment of material is to be purchased, a few samples 
should be taken from different parts of the supply. 

Fig. 195 shows a cylindrical and rectangular section 
of iron or steel as used in a tension test. The pieces 
are machined all over and filed along the middle part, 
for the body of the material should be as free from tool 
marks as possible and of a uniform size. The propor- 
tion of the parts is given in terms of the cross section of 
the material. 

The blocks shown in Fig. 196 represent about the 



STRENGTH OF MATERIALS 



217 



limit of the size of pieces used in compression tests. A 
piece should not be longer than 6 times its diameter, 
else a bending may occur along with the compression. 

The forms used for bending and shearing tests are 
hke those shown in Fig. 197, the form in each case de- 
pending on the kind of material and the result to be 
determined. For torsion tests, the specimens are hke 
those used for tension tests 




Fig. 196. 



^ 



a 



Fig. 197. 



83. Values of Ultimate Strengths. — From the results 
of tests with these machines the following table shows 
the average values of ultimate strengths of the most 
used materials and for the kind of stress indicated. 



Ultimate 


Strength in Pounds per Square Inch 


Material 


Tension Compression Shear 


Cast iron 

Wrought iron .... 
Average steel .... 

Soft copper 

Hard copper 

Wood 


20,000 

55,000 
60,000 
25,000 
40,000 
10,000 


90,000 
50,000 
70,000 
40,000 
55,000 
8,000 


20,000 
40,000 
50,000 
20,000 
40,000 
3,000 
(across grain) 



218 



MECHANICS AND ALLIED SUBJECTS 



The following results represent a fair average 



Material 


Tension 


Compressor 


Shear 


Granite 

Limestone 

Concrete 

Brick 


15,000 

12,000 

2,000 

5,000 


1,500 
1,400 
1,000 
1,000 



It would be of course unsafe to use in actual serviec 
material under such conditions that it would be subjected 
to a stress near the ultimate stress of the material, since 
the piece might break at any moment. A piece of 
material put in actual service is therefore designed for 
a stress per square inch, which is much less than the 
ultimate stress. This stress is called the working stress 
or the safe working stress, 

84. Factor of Safety. — The ultimate strength divided 
by the working stress is called the Factor of Safety. 
For example if a piece of steel breaks at 80,000 lb. per 
sq. in. and we use a factor of safety of 8, the working 
stress is only }i of 80,000 lb. or 10,000 lb. per sq. in., 
and the design is worked out for the latter value. The 
factor of safety of a material varies according to the 
material used and the manner of loading. A much 
larger factor of safety is allowed for bridge members on 
which there is a varying load than for a building for 
example where the load is practically constant. 

85. The Elastic Limit. — In designing structures such 
values of working stress are always taken that the 
material will not be stretched beyond its ^^ elastic limit, ^^ 
that is will not be worked at a unit stress beyond which 
the material will not return to its original shape when 
the stress is removed. The Elastic Limit which should 
always be greater than the working stress may also be 
defined as that unit stress beyond which the deforma- 



STRENGTH OF MATERIALS 



219 



tions of the material increase in a faster ratio than the 
appUed loads, and beyond which the material will ac- 
quire a permanent set on the removal of the load, and 
not return to its original shape. 

86. Values of Safe Working Stresses. — Following are 
safe working stresses in pounds per square inch for the 
most commonly used materials. The safe working load 
for tension, compression and shear are abbreviated re- 
spectively as follows Stj Scj Ss. 

Safe Working Stresses — Pounds per Square Inch 



Material 


Tension 

St 


Compression 


Shear 

Ss 


Cast iron 

Wrought iron. . . . 
Steel castings .... 
Timber 


4,000 

10,000 

12,000 

800 


12,000 

12,000 

16,000 

800 


3,000 

10,000 

12,000 

600 




(with grain) 



From the above we have the rule, the safe load ^^W^^ 
which can be carried for the several kinds of stresses is as 
follows : 

For tension W = area in sq. in. X &. 

For compression W = area in sq. in. X Sc. 
For shear W = area in sq. in. X Ss. 

Or in general for all stresses 



from which 



W = area X aS 
W 



area = 



S 



These mean that the total safe load equals the stress 
per square inch times the area of cross section of the 
piece in square inches. And also the area required by 



220 



MECHANICS AND ALLIED SUBJECTS 



a piece equals the total load divided by the safe load per 
square inch. 

From the calculated area, the length of a side of the 
cross section, or the diameter in case of a circle, and so 
forth may be found. 




Fig. 198. 



87. Strength of Rods. — When steam is applied to the 
back side of the piston shown in Fig. 198, the pressure 
puts the rod in tension. If the pressure is 180 lb. per 
sq. in. the total pressure on the piston is 180 times the 
net area of the piston, that is, subtracting the area of 

cross section of the rod. The area on 
which the pressure acts is shown cross 
sectioned in Fig. 199 and equals 

(24)2 X 735 _ (3 25)2 >< 735 ^j. 
444.1 sq. in. 

The total pull on the rod therefore 
equals 

444.1 X 180 or 79,900 lb. 

that is, the applied load is 79,900 lb. and the total resisting 
stress is therefore 79,900 lb. The stress per sq. in. on the 
rod equals 

total load 79,900 




area of rod 



or 



8.30 



STRENGTH OF MATERIALS 221 

which equals 9630 lb., and this is well within the safe 
limit for steel. The unit compressive stress is also 
within the safe limit for steel. The compressive stress 
occurs when steam is admitted to the head end of the 
cyhnder. 

Example. — Find the size of a circular steel rod from 
which a weight of 4 tons is to be suspended. 

Solution. — Taking 14,000 lb. per sq. in. as the safe 
working load in tension, we have 

W 4 X 2000 
Area = -^ or ... ^^^ or .572 sq. m. 

The diameter of the rod then equals 

Varea /.572 oro • 

In figuring out threaded bolts the required area must 
be taken as that at the root of the threads, since a piece 
is no stronger than its weakest part. If the bolt has a 
sufficient strength for the area at the root of its thread, or 
in other words for its smallest area, the body of the bolt 
will have a surplus of strength on account of its greater 
area of cross section. 

88. Strength of Ropes, Chains and Cables. — The ulti- 
mate strength of hemp rope is about 6000 lb. per sq. in. 
and for Manila rope about 3000 lb. per sq. in. 

A safe working stress for hemp rope is 1400 lb. per sq. 
in. based on the Nominal Area of the rope, that is, as- 
suming the area of cross section of the rope a solid circle. 
For Manila rope half the above value may be used. 

The safe working load for chain links is given as 9000 
lb. per sq. in. For iron cables the ultimate strength is 
about 40,000 lb. per sq. in. and for steel cables 80,000 



222 MECHANICS AND ALLIED SUBJECTS 

lb. per sq. in. A factor of safety of 10 is often used for 
ropes and cables in finding the safe working stresses. 
This makes the safe value of working stress for iron cable 
4000 lb. per sq. in. and for steel cable 8000 lb. per sq. in. 
89. The Strength of Columns. — When a bar or rod 
subjected to compression has a length greater than 10 
times its diameter it is called a column^ and it must be 
worked out by difficult formulas which take account of 
its length, size and form of cross section. Columns 
may fail by buckling as well as being crushed, and most 
satisfactory results in some cases are obtained by test- 
ing the actual columns in a suitable testing machine- 
From the results of such tests the most reliable informa- 
tion on the strength of the columns is obtained. 

PROBLEMS 

1. It took 100,000 lb. pull in a testing machine to break a steel 
rod lyz in. in diam. What was its ultimate tensional strength in 
lb. per sq. in.? 

2. If a cast-iron bar 1 J^ in. by 2 in. in cross section breaks under 
a tensile load of 70,000 lb., what load in lb. per sq. in. would break 
a 2\^-\n. diam. cast-iron rod of equal unit strength? 

3. A round cast-iron bar is to be subjected to a tension of 34,000 
lb. If it is designed so that the unit stress is 2500 lb. per sq. in., 
find the diameter of the rod. 

4. A wooden block 1% in. square was crushed in a testing 
machine under a load of 21,640 lb. What was its ultimate com- 
pressive strength in lb. per sq. in.? 

5. It required 87,630 lb. to shear off in a testing machine a 
iM-in. diam. steel rod. What was its ultimate shearing strength 
in lb. per sq. in.? 

6. If the ultimate compressive stress of wrought iron is 55,000 
lb. per sq. in. and it is subjected to a working stress of 11,500 lb. 
per sq. in., what is its factor of safety? 

7. A %-in. diam. wrought-iron tie rod on a bridge is subjected 
to a working stress of 9000 lb. per sq. in. What load is it sup- 



STRENGTH OF MATERIALS 



223 



porting? If the ultimate stress is 50,000 lb. per sq. in., what 
factor of safety is used in its design? 

8. Find the compression in lb. per sq. in.fora3-in. diam. piston 
rod carrying an 18-in. diam. piston with a steam pressure of 1801b. 
per sq. in. 

9. Find the tensile stress in lb. per sq. in. for a 3J^^-in. diam. 
piston rod with steam at 175 lb. per sq. in. acting on the back of 
a 20-in. diam. piston attached to the rod. 

(Note. — Subtract the cross-sectional area of the piston rod from 
the piston area to get the net area on which the steam acts.) 

10. What is the stress per sq. in. in the studs holding on a 
cylinder head for a 24-in. diam. cylinder if there are twenty-six 



< V^ - v^ 



} 



^ Shear 



A 



^I^ 



5 



Fig. 200. 



Fig. 201. 



%-m. diam. studs and the greatest steam pressure is 200 lb. 
per sq. in? 
• (Note.— The area at root of thread of each %-in. stud is .419 
sq. in.) 

11. A solid cast-iron cylinder 2>^ in. in diam. is under a com- 
pression of 40,000 lb. Find its factor of safety. 

12. A brick 2 in. X 4 in. X 8 in. weighs about 4>^ lb. Find the 
height of a pile of bricks so that the compressive stress per sq. in. 
on the lowest brick will be y% of the ultimate strength. 

13. A wrought-iron bolt as shown in Fig. 200 is to be sub- 
jected to shear in two places. If the load is 3000 lb. and the factor 
of safety 5, find the diameter of the bolt. 



224 MECHANICS AND ALLIED SUBJECTS 

14. What size of hemp rope is required to Hft a casting weighing 
2>^ tons? 

15. What is the total safe load which a steel cable 2 in. in 
diam. can stand? 

16. For an ultimate tensile strength of 50,000 lb. per sq. in. 
find the strength of the %-in. diam. cone rivet shown in Fig. 201. 



INDEX 



Acceleration of motion, defini- 
tion of, 75 
formula for, 76 
Applications of triangular func- 
tions to right tri- 
angles, 167 
to oblique triangles, 
186 



Current, calculation of elec- 
trical, 194 
Cutting speeds of lathes, 80 

Definition of forces, 1 
Density, 14 

Differential pulley, the, 51 
Division by logarithms, 121 



Belts and pulleys, 69 

length of, 70 
Belting, formula for, 101 

horsepower of, 100 
Bolt heads and nuts, rules for 

28 
Brake horsepower, 96 

Cables, strength of, 221 
Calculation of electrical power, 

197 
Center of gravity, definition of, 

7 
to find, 8 
Chains, strength of, 221 
Characteristic, the, 116 
Coefficients of expansion of 

solids (table), 110 
Columns, strength of, 222 
Composition of motions, 76 
Contraction due to heat, 108 
Cosine of an angle, the, 129 
Cotangent of an angle, the, 

129 



Efficiency of machines, 73 
Elastic limit, the, 218 
Electric current, calculation of, 
194 
heat produced by, 112 
resistance, calculation of, 

194 
voltage, calculation of, 194 
wires, safe carrying capac- 
ity of, 194 
Electrical energy, methods of 
charging for, 202 
horsepower, 200 
power, calculation of, 197 
resistances, calculation of, 
190 
in parallel, 192 
in series, 191 
Electricity, 190 
Energy, definition of, 102 
Expansion, coefficients of, of 
solids (table), 110 
due to heat, 108 
rule for amount of, 110 



225 



226 



INDEX 



Factor of safety, 218 
Force of gravity, 76 
Forces, definition of, 1 

parallel, 2 

passing through a point, 2 
Fulcrum of lever, 31 
Functions, of an angle, table of 
rales for, 130 

line values of, 157 

Gases, volume and pressure of, 

87 
Gears, definitions of, 59 
Gear trains, 60 

Gravity, action on falling ob- 
jects, 5 

definition of, 5 

force of, 76 

work done against, 6 
Grinding speeds for tools, 81 
Grind stones, speeds of, 81 

Heat, definition, 105 

expansion and contraction 

due to, 108 
latent, 112 
measurement, 107 
mechanical equivalent of, 

112 
produced by the electric 

current, 112 
specific. 111 
Horsepower, definition of, 91 
brake, 94, 96 
electrical, 200 
formula for cylinder, 93, 97 
of belting, 100 

Inclined plane, the, 53 



Kinds of stresses, 209 
Kinetic energy, 102 

Latent heat, 112 

Lathe, cutting speed of, 80 

gears, 61 
Levers, classes of, 31 

definition of, 31 

fulcrum of, 31 

moment of, 32 
Line values of functions, 157 
Logarithms, definition of, 116 

multiplication by, 121 

the characteristic of a, 116 

the mantissa of a, 116 

table, 117 

to divide by, 121 

to find number from a, 120 
roots by, 123 
powers by, 122 

Machines, efficiency of, 73 
Mantissa, the, 116, 119 
Materials, strength of, 208 
Measurement of heat, 107 

of right triangles, 127 
Mechanical equivalent of heat, 

112 
Moment of lever, 32 
Motions, acceleration of, 75 

composition of, 76 

definition of, 75 

formula for acceleration of, 
76 
Multiplication, by logarithms, 
121 

Oblique triangles, appli(;ations 
of triangular functions 
to, 186 



INDEX 



227 



Oblique triangles, methods . of 
working out, 177 

Ohm's law, 194 

Parallel forces, 2 
Pitch of thread, 20 
Potential energy, 103 
Power, calculation of electrical, 
197 

horse, 91 
Pressure and volume of gases, 

87 
Prony brake, 93 
Pulleys, 48, 69 

arrangement of, 48 

calculations for, 69 

differential, 51 

Resistances, calculation of elec- 
trical, 190, 194 
in parallel, 192 
in series, 191 
Resultant, the, 2 
Right triangles, definition, 127 
methods of working out, 

160 
rules for measurement of, 
127 
Rods, strength of, 220 
■Ropes, strength of, 221 
Rules fir functions of an angle, 
table of, 130 

Safe working stresses, values of, 

219 
Screw, cutting, lathe gears for, 

61, 64 
jack, the, 56 



Screw threads, 20 

construction of, 22, 23, 

26 
diameter at root of, 25 
lead of, 20 
pitch of, 20 
Sine of an angle, the, 129 
Specific gravity, 14 

heat. 111 
Speeds, cutting, 80 
definition of, 75 
grinding, 81 
of grindstones, 81 
Stability of objects, 12 
Standard test pieces, 216 
Strain, 209 

Strength, of cables, 221 
of chains, 221 
of columns, 222 
of materials, 208 
of rods, 220 
ultimate, 211 
Stresses, 209 

kinds of, 209 

Tables of functions, 133 

explanation of use of, 131 
of logarithms, 117 

use of, 119 
of rules for functions of an 
angle, 130 
Tangent of an angle, the, 129 
Temperature, 105 
Testing machines, 212 
Test pieces, 216 
Therniometer, 106 
Threads, construction of, 22, 23, 
26 
diameter at root of, 25 



228 



INDEX 



Threads, lead of, 20 

pitch of, 20 
Triangles, methods for working 
out right, 160 
out oblique, 177 
the right, 127 

rules for measurement 
of, 127 
Triangular functions, applica- 
tions of, 167 

Ultimate strength, 211 

strengths, values of, 217 
Use of tables of functions, ex- 
planation of, 131 



Velocity, definition of, 75 

formula for, 76 
Voltage, calculation of electric, 

194 
Volume and pressure of gases, 

87 



Wedge, the, 54 

Wires, safe carrying capacity of, 

194 
Work, definitions and rules on, 

91 
Working stresses, safe values of, 

219 



rx- 



